Intergenerational Transfers and the Distribution of Earnings

Glenn C Loury

Econometrica Vol 49 No 4 (Jul 1981) pp 843-867

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Econometrics Vol 49 No 4 (July 1981)

INTERGENERATIONAL TRANSFERS AND THE DISTRIBUTION OF EARNINGS

This paper models the dynamics of the earnings distribution among successive genera- tions of workers as a stochastic process The process arises from the random assignment of abilities to individuals by nature together with the utility maximizing bequest decisions of their parents A salient feature of the model is that parents cannot borrow to make human capital investments in their offspring Consequently the allocation of training resources among the young people of any generation depends upon the distribution of earnings among their parents This implies in turn that the often noted conflict between egalitarian redistributive policies and economic efficiency is mitigated A number of formal results are proven which illustrate this fact

1 INTRODUCTION

EXPLAININGTHE DISTRIBUTION OF INCOME is among the central tasks of econom- ics Indeed the classical query What determines the division of product among the factors land labor and capitalhas stimulated some of the most profound contributions to economic thought In modern neoclassical writing the size distribution of income particularly the distribution of wage income has become a topic of interest This topic has many facets not all of which can be studied theoretically within the same model The present paper focuses upon one empirically important aspect of the process by which a distribution of earnings capacities arises in a population-the consequences of social status or family background for individual earnings prospects A commonly observed fact is that earnings correlate positively across generations2 One explanation for which there is some evidence is that the acquisition in youth of productivity enhancing characteristics is positively affected by parental i n ~ o m e ~

Here we pursue this notion in some depth A choice theoretic framework is used to model the effect of parental income on offsprings productivity An economy is constructed with an elementary social structure Individuals who live for two periods are arranged into families Each family has one young person and one old person Think of the older family member as the parent and the younger as the offspring Families exist contemporaneously but are isolated from each other They each produce a perishable good in a quantity which depends on the ability and the training of the parent Each individual of this stationary population is assumed to learn during the second period of life the value of his randomly assigned ability endowment Each family divides its income of the

This paper derives from the authors PhD Thesis The advice of Professor Robert Solow is gratefully acknowledged and comments of Carl Futia Sanford Grossman and Joseph Stiglitz have also been helpful All errors are of course my own

2 ~ e e for example Jencks et al [15] see Duncan et al [lo] Datcher [8]and Jencks et al [15] A discussion of the consequences of

this observation for the analysis of racial income differences can be found in Loury [18 Ch I]

844 GLENN C LOURY

perishable commodity between consumption and training for the offspring Offsprings abilities independent and identically distributed random variables are unknown when this division is made

Parents control family resources They are motivated to forego consumption and invest in their offspring by an altruistic concern for the well-being of their progeny Because parents are of varying abilities and backgrounds their incomes differ Consequently the training investments which they make in their offspring also vary If the production function linking ability and training to output has diminishing returns to training then two parents making different income constrained investment decisions face divergent expected marginal returns to training in terms of their offsprings earnings If the parent investing less facing the higher expected marginal return could induce the parent investing more to transfer to his son a small amount of the others training resource then the offspring of both families could on average (through another inter-family income transfer in the opposite direction) have greater incomes in the next period If however such trades are impossible because the relevant markets have failed then the distribution of income among parents will affect the efficiency with which overall training resources are allocated across offspring (This will be true even when abilities are correlated within families across generations since parents of the same ability will also have different incomes) The model pre- sented here captures some features positive and normative of the situation which arises when such training loans between families are not possible

Because family income in a particular generation depends on the level of training which the parent received in the previous period and that training investment depended on the familys income in the preceding generation the distribution of income among a given generation depends on the distribution which obtained in the previous generation A stochastic process for the income distribution in this economy is thus implied by the parental decision-making calculus and the random assignment of ability It is natural to think of an equilibrium or stationary distribution as one which if it obtains persist^^ We demonstrate for our model the existence and global stability of such an equilib- rium distribution Moreover because parents are altruistic regarding their off- spring but uncertain about the earnings (and hence welfare) of their children institutional arrangements which redistribute income among subsequent genera- tions affect the wellbeing of the current generation Such redistributive arrange- ments can have insurance-like effects We demonstrate below that under certain conditions egalitarian redistributive measures can be designed which make all current members of society better off

The remainder of the paper is organized as follows In the next two sections the analytical model is described and the main mathematical results are stated Section 4 subjects the equilibrium distribution to more detailed examination and

4Becker and Tomes investigate a similar notion of equilibrium in [3] Varian [29] has discussed insurance aspects of redistributive taxation in a single period model

845 INTERGENERATIONAL TRANSFERS

a number of economically interesting results are obtained As mentioned risk spreading opportunities permitted by the independence of the random ability endowments across families are explored Moreover inefficiency in the allocation of training resources among the young as a consequence of incomplete loan markets is noted Public provision of training is shown under some conditions to increase output and reduce inequality The section concludes with an examina- tion of the joint distribution of ability and earnings in equilibrium Here we assess the extent to which reward may be presumed to be distributed according to merit We find the notion of meritocracy appropriately defined as difficult to defend when social background constrains individual opportunity In Section 5 we provide a complete solution of our model in the case of Cobb-Douglas utility and production functions and uniform ability distribution The equilibrium distribution is explicitly derived and studied Proofs of the various results appear in Section 6 while concluding observations are mentioned in the final section

2 THE BASIC MODEL

Imagine an economy with overlapping generations along the lines of Samuel- son [26]composed of a large number of individuals Time is measured discretely and each individual lives for exactly two periods An equal number of individu- als enter and leave the economy in each period implying a stationary popula- tion Every person in the first period of life (youth) is attached to a person in the second period of life (maturity) and this union is called a family

The family is the basic socioeconomic unit in this idealized world We take the mature member as family head making all economic decisions The income (output) of a family depends on the productivity of the head which in turn depends upon his training and innate ability6 Family income must be divided in each period between consumption and training of the youth We assume that such investment is the only means of transferring goods through time The training which a mature person possesses this period is simply that portion of family income invested in him last period

Production of the one perishable commodity is a family specific undertaking It requires no social interaction among families nor the use of factors of production other than the mature members time which is supplied inelastically Consumption is also a family activity with young and mature individuals deriving their personal consumption from the family aggregate in a manner determined by social custom and not examined here Natural economic ability differs among individuals each person beginning life with a random endowment of innate aptitudes Moreover the level of this endowment becomes known only in maturity Respectively denoting by x a and e a mature individuals output

6~bility here means all factors outside of the individuals control which affect his productive capacity

We abstract from physical capital and real property so as to focus on intergenerational transmission of earnings capacity

846 GLENN C LOURY

ability and training we write the production possibilities as

(1) x = h ( a e)

The technology h( ) is assumed common to all families in all periods The decision to divide income between consumption and training is taken by

mature people to maximize their expected utility We assume that all agents possess the same utility function The utility of a parent is assumed to depend on family consumption during his tenure and on the wellbeing which the offspring experiences after assuming family leadership9 This wellbeing is taken by parents to be some function of the offsprings future earnings Earnings of offspring for any level of training decided upon by the parent is a random variable determined by (1) and the distribution of a Call the function which parents use to relate the prospective income of their offspring to the childs happiness an indirect utility function Call an indirect utility function consistent if it correctly characterizes the relationship between maximized expected utility and earnings for a parent whenever it is taken to characterize that same relationship for the offspring That is an indirect utility function V is consistent if and only if V solves

[aygt 0 V ( y )= mar E U(c V ( h ( a y - c ) ) ) ] o lt c lt y

where U(c V ) is parental utility when family consumption is c and offspring expected utility is V and E is the expectation operator under the random distribution of innate ability Consistency is assured by (2) because expected utility maximization leads to an indirect utility function for mature individuals identical to that which they presume for their offspring Along with a solution for (2)will come an optimal investment function e(y) which relates parents income to the training of their offspring The properties of these functions are studied below

Notice for now that the earnings of a mature individual ( x ) depends on his random ability and parents earnings ( y )in the following way

Equation (3) is the basic relationship which characterizes social mobility in this economy It establishes a stochastic link between parents and offsprings earn-ings Each individual has limited mobility in that economic achievement varies with aptitude but in a manner determined by parental success Moreover the utility obtained from having a particular level of income V ( y ) depends

he assumption of identical tastes for everyone is purely for convenience It may be replaced with the hypothesis of inheritance of preferences within the family only without affecting any of the formal results

~ l t r u i s m is modelled in this way (ie with the cardinal utility of children entering the parents utility function) rather than with childs consumption in the parents utility function (as in Kohlberg [16]) to avoid the possibly inefficient growth paths associated with the latter set-up (see Diamond [9])

847 INTERGENERATIONAL TRANSFERS

(through equation (2)) on the extent of social mobility This is because inter- generational altruism as modelled here implies that parents are indirectly concerned about the incomes of all of their progeny Thus when there is a great deal of mobility (because a has a large variance andor ahaa is large) a low income will be less of a disadvantage and a high income less of an advantage than when there is very little social mobility Thus the graph of V() will be flatter when the society is more mobile This suggests that our indirect utility function may be useful for assessing the relative merits of social structures generating different degrees of social mobility across generations We shall return to this point below

Summarizing there are three distinct elements which determine the structure of our model economy the utility function the production function and the distribution of the ability endowment We require some assumptions concerning these elements which are informally described here First we shall assume that some consumption is always desirable parents are risk averse and parents care about but discount the wellbeing of their offspring Concerning production we assume diminishing returns to training a strictly positive marginal product of ability that the net marginal return to training eventually becomes negative for all individuals and the net average return is always negative for the least able Finally we assume an atomless distribution of ability with connected and bounded support we rule out autocorrelation in innate endowment^^ and make extreme values of a sufficiently likely that parents will always desire to invest something in their offspring The formal statement of these assumptions follows (R+ is the nonnegative half-line)

ASSUMPTION +R+ is a strictly increasing strictly concave twice 1 (i) U R continuously differentiable function on the interior of its domain satisbing U(0O) = 0 (ii) V V gt 0

lim -au (c V) = +KI ~~0 ac

(iii) There exists y gt 0 such that V(c V) E R (a Ua V) E [y 1 - y ]

ASSUMPTION +R+ is continuously differentiable strictly increasing 2 (i) h R strictly concave in e satisfving h(0O) = 0 and h(0 e) lt e Ve gt 0

ah(ii) - ( a e ) gt p gt O V(ae)euroRaa

OPositive intergenerational correlation of abilities would be a more natural assumption (For a discussion of the evidence see Goldberger [13])We have assumed this away for simplicitys sake One might think this affects the basic character of our argument since if rich parents (and hence their children) are smarter than poor parents it may not be inefficient for them to invest relatively more in their offspring But as our discussion of meritocracy in Section 4 illustrates when parental resources determine human capital investments it is not necessarily true that those with greater income have (probabilistically) greater ability

848 GLENN C LOURY

(iii) There exists p gt 0 I gt 0 such that

ah e gt I max -(a e) 9 p lt 1 a ~ [ o ae11

ASSUMPTION3 Innate economic ability is distributed on the unit interval inde- pendently and identically for all agents The distribution has a continuous strictly positive density function f [0 11 +R+

3 CONSISTENT INDIRECT UTILITY AND THE DYNAMICS OF THE INCOME

DISTRIBUTION

Our assumptions (hereafter Al-A3 etc) enable us to demonstrate the exist- ence of a unique consistent indirect utility function V(y) and to make rather strong statements about the long run behavior of the distribution of earnings implied by equation (3) These essentially mathematical results are required before we can proceed to the economic analysis which is developed in the following section To see that the notion of consistent indirect utility is not vacuous we adopt to our context a dynamic programming argument originated by Bellman First however notice that A2(iii) implies there is a unique earnings level y such that h(1J) = y and y gty h(1 y ) lty Hence we may without loss of generality restrict our attention to incomes in the interval [0 j ] since no familys income could persist outside this range Let C [0 j ] denote the continu- ous real valued functions on [0 y] For 9 E C[O j ] define the function T+() as follows

(4) vy euro [0 y] T+(y) Eo7a E U(c 9(h(a Y - ~ 1 ) ) c y

An indirect utility function is consistent if and only if it is a fixed point of the mapping T Our assumptions permit us to show that T C [0 J] C [0 j ] is a contraction mapping and hence has a unique fixed point In Section 6 we prove the following

THEOREM1 Under Al A2 and A3 for the above defined positive number j there exists a unique consistent indirect utility function V on [0 y] V is strictly increasing strictly concave and differentiable on (0 y] The optimal consumption function c(y) is continuous and satisfies 0 lt c(y) lty Vy E (0 y] (Denote e(y) =y - c(y))

The property of uniqueness established here is very important For neither positive nor normative analysis would our story carry much force were we required to select arbitrarily among several consistent representations of family preferences On the other hand we have assumed a cardinal representation of preferences This is unavoidable if parents are to be taken as contemplating the welfare of their offspring While an axiomatic justification of our choice criterion might be sought along the lines of Koopmans [17] this task is complicated by the

849 INTERGENERATIONAL TRANSFERS

uncertainty inherent in our set-up and is not pursued here As one might expect V inherits the property of risk aversion from U This is important for much of the discussion of Section 4

We turn now to the study of the intergenerational motion of the distribution of income In any period (say nth) the distribution of family income is conceived as a probability measure on [0 j ] v The interpretation is that for A c [0 J] vn(A) is the fraction of all mature agents whose incomes lie in the set A In what follows we denote by 9 the set of probability measures on 9the Bore1 sets of [0 j ] p will denote the Lebesgue measure on 9

The system of family decision making outlined above enables us to treat income dynamics in this economy as a Markoff process To do so we require the following definition

DEFINITION1 A transition probability on [0 j ] is a function P [0 j ] x +[O 11 satisfying (i) Vy E [0 j ] P(y a ) E 9 (ii) VA E P( A) is a -measurable function on R+ To each transition probability on [0 j ] there corresponds a Markoff process That is once a transition probability has been specified one can generate for any fixed v E9a sequence of measures v) as follows

Thus the intergenerational motion of the distribution of income will be fully characterized once we have found the mechanism which relates the probability distribution of an offsprings income to the earnings of its parent Towards this end define the function h - x R++ as follows

Then it is clear that the probability an offspring has income in A when parents earnings is y is simply the probability that the offspring has an innate endow- ment in the set hK1(~e(y)) The relevant transition probability is given by

With P defined as in (6) it is obvious that P(y ) E9moreover it follows from a result of ~ u t i a that P ( A) is -measurable when e() is continuous The last fact was shown in Theorem 1

We now introduce the notion of an equilibrium distribution

DEFINITION2 An equilibrium income distribution is a measure v E 9 satisfy-ing VA E

See Futia [12 Theorem 52 p 741

850 GLENN C LOURY

Thus an equilibrium distribution is one which if it ever characterizes the distribution of earnings among a given generation continues to do so for each subsequent generation Assumptions A1-A3 already permit us to assert the existence of at least one equilibrium and convergence to some equilibrium However since our focus here is on dynamics it seems preferable to introduce the following simple assumption from which follows the global stability of the unique equilibrium point

That is education is a normal good We now have the following theorem

THEOREM2 Under Al-A4 there exists a unique equilibrium distribution v E 9If v) is a sequence of income distributions originating from the arbitrary initial v then limnvn = v pointwise on 9Moreover v has support [OA where y satisfies h (1 e($) =y

Thus by the proof given in Section 6 we establish existence uniqueness and stability for our notion of equilibrium Together with Theorem 1 this constitutes the framework within which we explore the issues raised in the introduction This examination is conducted in the next two sections

4 ECONOMIC ANALYSIS OF THE EQUILIBRIUM DISTRIBUTION

We have now constructed a relatively simple maximizing model of the distribu- tion of lifetime earnings among successive generations of workers The salient features of this model are (i) the recursive family preference structure reflecting intergenerational altruism (ii) the capital-theoretic characterization of produc- tion possibilities (iii) the stochastic nature of innate endowments assumed independent within each family over time and across families at any moment of time and (iv) the assumed absence of markets for human capital loans and income risk sharing between families These factors determine the average level of output in equilibrium and its dispersion as well as the criterion which families use to evaluate their wellbeing

Thus we may think of our model as defining a function M which associates with every triple of utility production and ability distribution functions satisfy- ing our assumptions a unique pair the equilibrium earnings distribution and consistent indirect utility

M (Uh f ) -+(v V)

A natural way to proceed in the analysis of this model is to ask comparative statics questions regarding how the endogenous (v V) vary with changes in the exogenous (U h f) This is a very difficult question in general though we are able to find some preliminary results along these lines presented below

Before proceeding to this however let us notice how the model distinguishes the phenomenon of social mobility across generations from that of inequality in

INTERGENERATIONAL TRANSFERS 851

earnings within a generation The relationship between these alternative notions of social stratification has been the focus of some attention in the literature [4 271 Here the two notions are inextricably intertwined intergenerational social mobil- ity is a property of the transition probability P while cross-sectional inequality is (asymptotically) a property of the equilibrium distribution to which P gives rise The basic problem for a normative analysis of economic inequality raised by this joint determination is that the concept of a more equal income distribution is not easily defined For instance how is one to compare a situation in which there is only a slight degree of inequality among families in any given generation but no mobility within families across generations with a circumstance in which there is substantial intragenerational income dispersion but also a large degree of intergenerational mobility Which situation evidences less inequality In the first case such inequality as exists between families lasts forever while in the latter case family members of a particular cohort experience significant earnings differences but no family is permanently assigned to the bottom or top of the earnings hierarchy

The formulation put forward here provides a way of thinking about this trade-off Our equilibrium notion depicts the intragenerational dispersion in earnings which would persist in the long run On the other hand the consistent indirect utility function incorporates into the valuation which individuals affix to any particular location in the income hierarchy the consequences of subsequent social mobility As noted earlier a great deal of intergenerational mobility implies a relatively flat indirect utility function making the cross-sectional distribution of welfare in equilibrium less unequal than would be the case with little or no mobility Thus by examining the distribution in equilibrium of consistent indirect utility (instead of income) one may simultaneously account for both kinds of inequality within a unified framework Indeed the statistic

is a natural extension of the standard utilitarian criterion and provides a complete ordering of social structures (U h f)satisfying A1-A4

Let us consider now the absence of loan and insurance markets in this model These market imperfections have important allocative consequences A number of writers have called attention to the fact that efficient resource allocation requires that children in low-income families should not be restricted by limited parental resources in their access to training For example Arthur Okun has remarked [22 p 80-811 concerning the modern US economy that The most important consequence (of an imperfect loan market) is the inadequate development of the human resources of the children of poor families-which would judge is one of the most serious inefficiencies of the American Economy today (Emphasis added) While it is certainly the case that in most societies a variety of devices exist for overcoming this problem they are far from perfect For example early childhood investments in nutrition or pre-school education are fundamentally income-constrained Nor should we expect a competitive loan

I

852 GLENN C LOURY

market to completely eliminate the dispersion in expected rates of return to training across families which arises in this model Legally poor parents will not be able to constrain their children to honor debts incurred on their behalf Nor will the newly matured children of wealthy families be able to attach the (human) assets of their less well-off counterparts should the latter decide for whatever reasons not to repay their loans (Default has been a pervasive problem with government guaranteed educational loan programs which would not exist absent public underwriting) Moreover the ability to make use of human capital is unknown even to the borrower Thus genuine bankruptcies can be expected as people mature to learn they are not as able as their parents gambled they would be The absence of inter-family loans in this model reflects an important feature of reality the allocative implications of which deserve study

Another critical feature of this story is that parents bear theoretically insurable risk when investing in their offspring Earnings of children are random here because the economic ability of the child becomes known only in maturity Yet a significant spreading of risk is technologically possible for a group of parents investing at the same level since their childrens earnings are independent identically distributed random variables Again we may expect that markets will fail to bring about complete risk spreading the moral hazard problems of income insurance contracts are o b v i ~ u s ~

Any attempt by a central authority to correct for these market imperfections must cope with the same incentive problems which limit the scope of the competitive mechanism here Efforts by a government to alter the allocation of training among young people will affect the consumption-investment decision of parents Moreover parents may be presumed to have better information than a central authority about the distribution of their childrens abilities (This is especially so if ability is positively correlated across generations of the same family) Yet as long as a group of parents with the same information have different incomes and there is no capital market their investment decisions will vary causing inefficiency This inefficiency could be reduced if the government were to redistribute incomes more equally but only at the cost of the excess burden accompanying such a redistributive tax

However because we have assumed altruism between generations a program of income redistribution which is imposed permanently has welfare effects here not encountered in the usual tax analysis The fact that income will be redistri- buted by taxing the next generation of workers causes each parent to regard his offsprings random income as less risky Since parents are risk averse egalitarian redistributive measures have certain welfare enhancing insurance characteris-tics It may be shown that as long as the redistributive activity is not too extensive this insurance effect dominates the excess burden effect Under such circumstances a permanent redistributive tax policy can be designed in such a way as to make all current mature agents better off

See Zeckhauser [30]for a discussion of these problems

853 INTERGENERATIONAL TRANSFERS

DEFINITION3 An education specific tax policy (ESTP) is a function r R + R+ with the interpretation that r(ye) is the after-tax income of someone whose income is y and training is e An ESTP is admissible if h(a e) = r(h(a e) e) satisfies Assumption 2 An admissible ESTP is purely redistributive (PRESTP) if j[r(h(a e) e) - h ( a e)]f(a)p (da) = 0 for every e gt 0 Denote by C the class of admissible PRESTP C is a non-empty convex set

DEFINITION4 The partial ordering is more egalitarian than denoted + is defined on C by the requirement 7 3 7 if and only if for each e the after tax distribution of income under 7 is more risky in the sense of second order stochastic dominance (see for example [24]) than the after tax distribution under

These two definitions allow us to formalize the discussion above By a result of Atkinson [I] 7 3 7 implies the Lorenze curve of the after tax income distribu- tion for persons of a given level of training under 7 lies on or above the corresponding curve under 7 However since the investment function e() will depend on 7 the same statement will not generally hold for the overall distribu- tions of income Thus we must use the term more egalitarian with some caution Let (y e) E jh(a e)f(a)p (da) and ~ ( y e) = y for all (y e) E

2+relative to Crepresenting perfect income insurance is maximal in Then on the other hand represents laisset faire For B E (0 I ) let r = B7 + (1 - )I Then r E C Finally for every r E C let V denote the unique solution for (2) when h is replaced by h

THEOREM3 Under A1-A3 7 $ r2 V(y) gt V(y) Vy gt 0 T1T2E C

Theorem 3 states that the use of a more egalitarian PRESTP among future generations of workers will lead to a higher level of welfare for each mature individual currently The careful reader will notice that this result depends upon our assumption that labor supply is exogenous In general we may expect that an income tax will distort both the educational investment decisions of parents and their choices concerning work effort By allowing the income tax function to vary with the level of educational investment (redistributing income within education classes only) we avoid the negative welfare consequences of the former distor- tion It is possible to preserve the result in Theorem 3 in the face of the latter kind of distortion as well if we restrict our attention to small redistributive efforts To see this imagine that h(ae) determines output per unit of time worked that each parent is endowed with one unit of time and that parents utility depends on work effort as well as consumption and offsprings welfare Under these slightly more general circumstances one can repeat the development of Theorems 1 and 2 with the natural modifications that indirect utility becomes a function of output per unit time (instead of total income) and income is the product of work effort and h(a e) Now consider introducing the tax scheme r (described above) for small B For fixed labor supply it may be shown that

854 GLENN C LOURY

welfare at any level of income is a concave function of B Moreover Theorem 3 implies that welfare is an increasing function of B However from standard tax theory we know that the excess burden of the tax T~ is zero when B = 0 and has derivative with respect to equal to zero at B = 0 (That is the excess burden of a small tax is a second order effect) Thus for sufficiently close to zero the insurance effect described in Theorem 3 will outweigh the excess burden effect of the tax scheme leading to an increase in wellbeing for all parents of the current generation

This result suggests that the standard tradeoff between equity and efficiency is somewhat less severe in the second-best world where opportunities to acquire training vary with parental resources It is sometimes possible to design policies which increase both equity and efficiency As a further indication of this last point let us examine the impact of public education in this model By public education we mean the centralized provision of training on an equal basis to every young person in the economy For the purposes of comparison we contrast the laissez faire outcome with that which occurs under public education with a per-capita budget equal to the expenditure of the average family in the no-intervention equilibrium We assume for this result that education is financed with non-distortionary tax

ASSUMPTION is convex in y5 h(ae(y)) is concave in y and ah(ae(y))aa

Assumption 5 posits that the marginal product of social background declines as social background (ie parents income) improves but declines less rapidly for the more able We have proven in Section 6 the following result

THEOREM4 Under A1-A5 universal public education with a per-capita budget equal to the investment of the average family in the laissez faire equilibrium will produce an earnings distribution with lower variance and higher mean than that which obtains without intervention

Finally let us turn to an examination of the relationship between earnings and ability in equilibrium It is widely held that differences in ability provide ethical grounds for differences in rewards An often cited justification for the market determined distribution of economic advantage is that it gives greater rewards to those of greater abilities In our model such is not literally the case because opportunities for training vary with social origin Thus rewards correspond to productivity but not ability Productivity depends both on ability and on social background Nonetheless some sort of systematic positive relationship between earnings and ability might still be sought Imagine that we observe in equilibrium an economy of the type studied here Suppose that we can observe an agents income but neither his ability nor his parents income Now the pair (iux) of individual ability and income is a jointly random vector in the population A very weak definition of a meritocratic distribution of income is that iu and x are positively correlated Under this definition the distribution is meritocratic if a regression of income on ability would yield a positive coefficient in a large

855 INTERGENERATIONAL TRANSFERS

sample A more substantive restriction would require that if x gt x then pr[G gt a 1 x = x ] gt pr[G gt a I x = x2]for all a E [0 11 Under this stronger defi- nition of a meritocratic distribution of rewards we require that greater earnings imply a greater conditional probability that innate aptitude exceeds any pre- specified level The following theorem asserts that the economy studied here is generally weakly meritocratic but not strongly so

THEOREM5 Under A1-A4 the economy is weakly meritocratic However it is not generally true that the economy is strongly meritocratic

The reason that a strong meritocracy may fail to obtain can be easily seen Suppose there exists some range of parental incomes over which the investment schedule e(y) increases rapidly such that the least able offspring of parents at the top earn nearly as much as the most able offspring of parents at the bottom of this range Then the conditional ability distribution of those earning incomes slightly greater than that earned by the most able children of the least well-off parents may be dominated by the conditional distribution of ability of those whose incomes are slightly less than the earnings of the least able children of the most well-off parents An example along these lines is discussed in Section 6

5 AN EXAMPLE

It is interesting to examine the explicit solution of this problem in the special case of Cobb-Douglas production and utility functions For the analysis of this section we assume that

h ( a e ) = (ae)

f ( a ) 3 1 a E [0 1 1

With these particular functions it is not hard to see that the consistent indirect utility function and optimal investment function have the following forms

(10) V ( y ) = k y s k gt 0 0 lt 6 lt 1 and

e (y ) = sy 0 lt s lt 1

The reader may verify by direct substitution that the functions V and e given above satisfy (2)when s = ( 1 + y)2 6 = 2 y ( l + y) and

By Theorem 1 we know these solutions are unique Thus in this simple instance we obtain a constant proportional investment

function This result continues to hold for arbitrary distributions of ability f and

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

844 GLENN C LOURY

perishable commodity between consumption and training for the offspring Offsprings abilities independent and identically distributed random variables are unknown when this division is made

Parents control family resources They are motivated to forego consumption and invest in their offspring by an altruistic concern for the well-being of their progeny Because parents are of varying abilities and backgrounds their incomes differ Consequently the training investments which they make in their offspring also vary If the production function linking ability and training to output has diminishing returns to training then two parents making different income constrained investment decisions face divergent expected marginal returns to training in terms of their offsprings earnings If the parent investing less facing the higher expected marginal return could induce the parent investing more to transfer to his son a small amount of the others training resource then the offspring of both families could on average (through another inter-family income transfer in the opposite direction) have greater incomes in the next period If however such trades are impossible because the relevant markets have failed then the distribution of income among parents will affect the efficiency with which overall training resources are allocated across offspring (This will be true even when abilities are correlated within families across generations since parents of the same ability will also have different incomes) The model pre- sented here captures some features positive and normative of the situation which arises when such training loans between families are not possible

Because family income in a particular generation depends on the level of training which the parent received in the previous period and that training investment depended on the familys income in the preceding generation the distribution of income among a given generation depends on the distribution which obtained in the previous generation A stochastic process for the income distribution in this economy is thus implied by the parental decision-making calculus and the random assignment of ability It is natural to think of an equilibrium or stationary distribution as one which if it obtains persist^^ We demonstrate for our model the existence and global stability of such an equilib- rium distribution Moreover because parents are altruistic regarding their off- spring but uncertain about the earnings (and hence welfare) of their children institutional arrangements which redistribute income among subsequent genera- tions affect the wellbeing of the current generation Such redistributive arrange- ments can have insurance-like effects We demonstrate below that under certain conditions egalitarian redistributive measures can be designed which make all current members of society better off

The remainder of the paper is organized as follows In the next two sections the analytical model is described and the main mathematical results are stated Section 4 subjects the equilibrium distribution to more detailed examination and

4Becker and Tomes investigate a similar notion of equilibrium in [3] Varian [29] has discussed insurance aspects of redistributive taxation in a single period model

845 INTERGENERATIONAL TRANSFERS

a number of economically interesting results are obtained As mentioned risk spreading opportunities permitted by the independence of the random ability endowments across families are explored Moreover inefficiency in the allocation of training resources among the young as a consequence of incomplete loan markets is noted Public provision of training is shown under some conditions to increase output and reduce inequality The section concludes with an examina- tion of the joint distribution of ability and earnings in equilibrium Here we assess the extent to which reward may be presumed to be distributed according to merit We find the notion of meritocracy appropriately defined as difficult to defend when social background constrains individual opportunity In Section 5 we provide a complete solution of our model in the case of Cobb-Douglas utility and production functions and uniform ability distribution The equilibrium distribution is explicitly derived and studied Proofs of the various results appear in Section 6 while concluding observations are mentioned in the final section

2 THE BASIC MODEL

Imagine an economy with overlapping generations along the lines of Samuel- son [26]composed of a large number of individuals Time is measured discretely and each individual lives for exactly two periods An equal number of individu- als enter and leave the economy in each period implying a stationary popula- tion Every person in the first period of life (youth) is attached to a person in the second period of life (maturity) and this union is called a family

The family is the basic socioeconomic unit in this idealized world We take the mature member as family head making all economic decisions The income (output) of a family depends on the productivity of the head which in turn depends upon his training and innate ability6 Family income must be divided in each period between consumption and training of the youth We assume that such investment is the only means of transferring goods through time The training which a mature person possesses this period is simply that portion of family income invested in him last period

Production of the one perishable commodity is a family specific undertaking It requires no social interaction among families nor the use of factors of production other than the mature members time which is supplied inelastically Consumption is also a family activity with young and mature individuals deriving their personal consumption from the family aggregate in a manner determined by social custom and not examined here Natural economic ability differs among individuals each person beginning life with a random endowment of innate aptitudes Moreover the level of this endowment becomes known only in maturity Respectively denoting by x a and e a mature individuals output

6~bility here means all factors outside of the individuals control which affect his productive capacity

We abstract from physical capital and real property so as to focus on intergenerational transmission of earnings capacity

846 GLENN C LOURY

ability and training we write the production possibilities as

(1) x = h ( a e)

The technology h( ) is assumed common to all families in all periods The decision to divide income between consumption and training is taken by

mature people to maximize their expected utility We assume that all agents possess the same utility function The utility of a parent is assumed to depend on family consumption during his tenure and on the wellbeing which the offspring experiences after assuming family leadership9 This wellbeing is taken by parents to be some function of the offsprings future earnings Earnings of offspring for any level of training decided upon by the parent is a random variable determined by (1) and the distribution of a Call the function which parents use to relate the prospective income of their offspring to the childs happiness an indirect utility function Call an indirect utility function consistent if it correctly characterizes the relationship between maximized expected utility and earnings for a parent whenever it is taken to characterize that same relationship for the offspring That is an indirect utility function V is consistent if and only if V solves

[aygt 0 V ( y )= mar E U(c V ( h ( a y - c ) ) ) ] o lt c lt y

where U(c V ) is parental utility when family consumption is c and offspring expected utility is V and E is the expectation operator under the random distribution of innate ability Consistency is assured by (2) because expected utility maximization leads to an indirect utility function for mature individuals identical to that which they presume for their offspring Along with a solution for (2)will come an optimal investment function e(y) which relates parents income to the training of their offspring The properties of these functions are studied below

Notice for now that the earnings of a mature individual ( x ) depends on his random ability and parents earnings ( y )in the following way

Equation (3) is the basic relationship which characterizes social mobility in this economy It establishes a stochastic link between parents and offsprings earn-ings Each individual has limited mobility in that economic achievement varies with aptitude but in a manner determined by parental success Moreover the utility obtained from having a particular level of income V ( y ) depends

he assumption of identical tastes for everyone is purely for convenience It may be replaced with the hypothesis of inheritance of preferences within the family only without affecting any of the formal results

~ l t r u i s m is modelled in this way (ie with the cardinal utility of children entering the parents utility function) rather than with childs consumption in the parents utility function (as in Kohlberg [16]) to avoid the possibly inefficient growth paths associated with the latter set-up (see Diamond [9])

847 INTERGENERATIONAL TRANSFERS

(through equation (2)) on the extent of social mobility This is because inter- generational altruism as modelled here implies that parents are indirectly concerned about the incomes of all of their progeny Thus when there is a great deal of mobility (because a has a large variance andor ahaa is large) a low income will be less of a disadvantage and a high income less of an advantage than when there is very little social mobility Thus the graph of V() will be flatter when the society is more mobile This suggests that our indirect utility function may be useful for assessing the relative merits of social structures generating different degrees of social mobility across generations We shall return to this point below

Summarizing there are three distinct elements which determine the structure of our model economy the utility function the production function and the distribution of the ability endowment We require some assumptions concerning these elements which are informally described here First we shall assume that some consumption is always desirable parents are risk averse and parents care about but discount the wellbeing of their offspring Concerning production we assume diminishing returns to training a strictly positive marginal product of ability that the net marginal return to training eventually becomes negative for all individuals and the net average return is always negative for the least able Finally we assume an atomless distribution of ability with connected and bounded support we rule out autocorrelation in innate endowment^^ and make extreme values of a sufficiently likely that parents will always desire to invest something in their offspring The formal statement of these assumptions follows (R+ is the nonnegative half-line)

ASSUMPTION +R+ is a strictly increasing strictly concave twice 1 (i) U R continuously differentiable function on the interior of its domain satisbing U(0O) = 0 (ii) V V gt 0

lim -au (c V) = +KI ~~0 ac

(iii) There exists y gt 0 such that V(c V) E R (a Ua V) E [y 1 - y ]

ASSUMPTION +R+ is continuously differentiable strictly increasing 2 (i) h R strictly concave in e satisfving h(0O) = 0 and h(0 e) lt e Ve gt 0

ah(ii) - ( a e ) gt p gt O V(ae)euroRaa

OPositive intergenerational correlation of abilities would be a more natural assumption (For a discussion of the evidence see Goldberger [13])We have assumed this away for simplicitys sake One might think this affects the basic character of our argument since if rich parents (and hence their children) are smarter than poor parents it may not be inefficient for them to invest relatively more in their offspring But as our discussion of meritocracy in Section 4 illustrates when parental resources determine human capital investments it is not necessarily true that those with greater income have (probabilistically) greater ability

848 GLENN C LOURY

(iii) There exists p gt 0 I gt 0 such that

ah e gt I max -(a e) 9 p lt 1 a ~ [ o ae11

ASSUMPTION3 Innate economic ability is distributed on the unit interval inde- pendently and identically for all agents The distribution has a continuous strictly positive density function f [0 11 +R+

3 CONSISTENT INDIRECT UTILITY AND THE DYNAMICS OF THE INCOME

DISTRIBUTION

Our assumptions (hereafter Al-A3 etc) enable us to demonstrate the exist- ence of a unique consistent indirect utility function V(y) and to make rather strong statements about the long run behavior of the distribution of earnings implied by equation (3) These essentially mathematical results are required before we can proceed to the economic analysis which is developed in the following section To see that the notion of consistent indirect utility is not vacuous we adopt to our context a dynamic programming argument originated by Bellman First however notice that A2(iii) implies there is a unique earnings level y such that h(1J) = y and y gty h(1 y ) lty Hence we may without loss of generality restrict our attention to incomes in the interval [0 j ] since no familys income could persist outside this range Let C [0 j ] denote the continu- ous real valued functions on [0 y] For 9 E C[O j ] define the function T+() as follows

(4) vy euro [0 y] T+(y) Eo7a E U(c 9(h(a Y - ~ 1 ) ) c y

An indirect utility function is consistent if and only if it is a fixed point of the mapping T Our assumptions permit us to show that T C [0 J] C [0 j ] is a contraction mapping and hence has a unique fixed point In Section 6 we prove the following

THEOREM1 Under Al A2 and A3 for the above defined positive number j there exists a unique consistent indirect utility function V on [0 y] V is strictly increasing strictly concave and differentiable on (0 y] The optimal consumption function c(y) is continuous and satisfies 0 lt c(y) lty Vy E (0 y] (Denote e(y) =y - c(y))

The property of uniqueness established here is very important For neither positive nor normative analysis would our story carry much force were we required to select arbitrarily among several consistent representations of family preferences On the other hand we have assumed a cardinal representation of preferences This is unavoidable if parents are to be taken as contemplating the welfare of their offspring While an axiomatic justification of our choice criterion might be sought along the lines of Koopmans [17] this task is complicated by the

849 INTERGENERATIONAL TRANSFERS

uncertainty inherent in our set-up and is not pursued here As one might expect V inherits the property of risk aversion from U This is important for much of the discussion of Section 4

We turn now to the study of the intergenerational motion of the distribution of income In any period (say nth) the distribution of family income is conceived as a probability measure on [0 j ] v The interpretation is that for A c [0 J] vn(A) is the fraction of all mature agents whose incomes lie in the set A In what follows we denote by 9 the set of probability measures on 9the Bore1 sets of [0 j ] p will denote the Lebesgue measure on 9

The system of family decision making outlined above enables us to treat income dynamics in this economy as a Markoff process To do so we require the following definition

DEFINITION1 A transition probability on [0 j ] is a function P [0 j ] x +[O 11 satisfying (i) Vy E [0 j ] P(y a ) E 9 (ii) VA E P( A) is a -measurable function on R+ To each transition probability on [0 j ] there corresponds a Markoff process That is once a transition probability has been specified one can generate for any fixed v E9a sequence of measures v) as follows

Thus the intergenerational motion of the distribution of income will be fully characterized once we have found the mechanism which relates the probability distribution of an offsprings income to the earnings of its parent Towards this end define the function h - x R++ as follows

Then it is clear that the probability an offspring has income in A when parents earnings is y is simply the probability that the offspring has an innate endow- ment in the set hK1(~e(y)) The relevant transition probability is given by

With P defined as in (6) it is obvious that P(y ) E9moreover it follows from a result of ~ u t i a that P ( A) is -measurable when e() is continuous The last fact was shown in Theorem 1

We now introduce the notion of an equilibrium distribution

DEFINITION2 An equilibrium income distribution is a measure v E 9 satisfy-ing VA E

See Futia [12 Theorem 52 p 741

850 GLENN C LOURY

Thus an equilibrium distribution is one which if it ever characterizes the distribution of earnings among a given generation continues to do so for each subsequent generation Assumptions A1-A3 already permit us to assert the existence of at least one equilibrium and convergence to some equilibrium However since our focus here is on dynamics it seems preferable to introduce the following simple assumption from which follows the global stability of the unique equilibrium point

That is education is a normal good We now have the following theorem

THEOREM2 Under Al-A4 there exists a unique equilibrium distribution v E 9If v) is a sequence of income distributions originating from the arbitrary initial v then limnvn = v pointwise on 9Moreover v has support [OA where y satisfies h (1 e($) =y

Thus by the proof given in Section 6 we establish existence uniqueness and stability for our notion of equilibrium Together with Theorem 1 this constitutes the framework within which we explore the issues raised in the introduction This examination is conducted in the next two sections

4 ECONOMIC ANALYSIS OF THE EQUILIBRIUM DISTRIBUTION

We have now constructed a relatively simple maximizing model of the distribu- tion of lifetime earnings among successive generations of workers The salient features of this model are (i) the recursive family preference structure reflecting intergenerational altruism (ii) the capital-theoretic characterization of produc- tion possibilities (iii) the stochastic nature of innate endowments assumed independent within each family over time and across families at any moment of time and (iv) the assumed absence of markets for human capital loans and income risk sharing between families These factors determine the average level of output in equilibrium and its dispersion as well as the criterion which families use to evaluate their wellbeing

Thus we may think of our model as defining a function M which associates with every triple of utility production and ability distribution functions satisfy- ing our assumptions a unique pair the equilibrium earnings distribution and consistent indirect utility

M (Uh f ) -+(v V)

A natural way to proceed in the analysis of this model is to ask comparative statics questions regarding how the endogenous (v V) vary with changes in the exogenous (U h f) This is a very difficult question in general though we are able to find some preliminary results along these lines presented below

Before proceeding to this however let us notice how the model distinguishes the phenomenon of social mobility across generations from that of inequality in

INTERGENERATIONAL TRANSFERS 851

earnings within a generation The relationship between these alternative notions of social stratification has been the focus of some attention in the literature [4 271 Here the two notions are inextricably intertwined intergenerational social mobil- ity is a property of the transition probability P while cross-sectional inequality is (asymptotically) a property of the equilibrium distribution to which P gives rise The basic problem for a normative analysis of economic inequality raised by this joint determination is that the concept of a more equal income distribution is not easily defined For instance how is one to compare a situation in which there is only a slight degree of inequality among families in any given generation but no mobility within families across generations with a circumstance in which there is substantial intragenerational income dispersion but also a large degree of intergenerational mobility Which situation evidences less inequality In the first case such inequality as exists between families lasts forever while in the latter case family members of a particular cohort experience significant earnings differences but no family is permanently assigned to the bottom or top of the earnings hierarchy

The formulation put forward here provides a way of thinking about this trade-off Our equilibrium notion depicts the intragenerational dispersion in earnings which would persist in the long run On the other hand the consistent indirect utility function incorporates into the valuation which individuals affix to any particular location in the income hierarchy the consequences of subsequent social mobility As noted earlier a great deal of intergenerational mobility implies a relatively flat indirect utility function making the cross-sectional distribution of welfare in equilibrium less unequal than would be the case with little or no mobility Thus by examining the distribution in equilibrium of consistent indirect utility (instead of income) one may simultaneously account for both kinds of inequality within a unified framework Indeed the statistic

is a natural extension of the standard utilitarian criterion and provides a complete ordering of social structures (U h f)satisfying A1-A4

Let us consider now the absence of loan and insurance markets in this model These market imperfections have important allocative consequences A number of writers have called attention to the fact that efficient resource allocation requires that children in low-income families should not be restricted by limited parental resources in their access to training For example Arthur Okun has remarked [22 p 80-811 concerning the modern US economy that The most important consequence (of an imperfect loan market) is the inadequate development of the human resources of the children of poor families-which would judge is one of the most serious inefficiencies of the American Economy today (Emphasis added) While it is certainly the case that in most societies a variety of devices exist for overcoming this problem they are far from perfect For example early childhood investments in nutrition or pre-school education are fundamentally income-constrained Nor should we expect a competitive loan

I

852 GLENN C LOURY

market to completely eliminate the dispersion in expected rates of return to training across families which arises in this model Legally poor parents will not be able to constrain their children to honor debts incurred on their behalf Nor will the newly matured children of wealthy families be able to attach the (human) assets of their less well-off counterparts should the latter decide for whatever reasons not to repay their loans (Default has been a pervasive problem with government guaranteed educational loan programs which would not exist absent public underwriting) Moreover the ability to make use of human capital is unknown even to the borrower Thus genuine bankruptcies can be expected as people mature to learn they are not as able as their parents gambled they would be The absence of inter-family loans in this model reflects an important feature of reality the allocative implications of which deserve study

Another critical feature of this story is that parents bear theoretically insurable risk when investing in their offspring Earnings of children are random here because the economic ability of the child becomes known only in maturity Yet a significant spreading of risk is technologically possible for a group of parents investing at the same level since their childrens earnings are independent identically distributed random variables Again we may expect that markets will fail to bring about complete risk spreading the moral hazard problems of income insurance contracts are o b v i ~ u s ~

Any attempt by a central authority to correct for these market imperfections must cope with the same incentive problems which limit the scope of the competitive mechanism here Efforts by a government to alter the allocation of training among young people will affect the consumption-investment decision of parents Moreover parents may be presumed to have better information than a central authority about the distribution of their childrens abilities (This is especially so if ability is positively correlated across generations of the same family) Yet as long as a group of parents with the same information have different incomes and there is no capital market their investment decisions will vary causing inefficiency This inefficiency could be reduced if the government were to redistribute incomes more equally but only at the cost of the excess burden accompanying such a redistributive tax

However because we have assumed altruism between generations a program of income redistribution which is imposed permanently has welfare effects here not encountered in the usual tax analysis The fact that income will be redistri- buted by taxing the next generation of workers causes each parent to regard his offsprings random income as less risky Since parents are risk averse egalitarian redistributive measures have certain welfare enhancing insurance characteris-tics It may be shown that as long as the redistributive activity is not too extensive this insurance effect dominates the excess burden effect Under such circumstances a permanent redistributive tax policy can be designed in such a way as to make all current mature agents better off

See Zeckhauser [30]for a discussion of these problems

853 INTERGENERATIONAL TRANSFERS

DEFINITION3 An education specific tax policy (ESTP) is a function r R + R+ with the interpretation that r(ye) is the after-tax income of someone whose income is y and training is e An ESTP is admissible if h(a e) = r(h(a e) e) satisfies Assumption 2 An admissible ESTP is purely redistributive (PRESTP) if j[r(h(a e) e) - h ( a e)]f(a)p (da) = 0 for every e gt 0 Denote by C the class of admissible PRESTP C is a non-empty convex set

DEFINITION4 The partial ordering is more egalitarian than denoted + is defined on C by the requirement 7 3 7 if and only if for each e the after tax distribution of income under 7 is more risky in the sense of second order stochastic dominance (see for example [24]) than the after tax distribution under

These two definitions allow us to formalize the discussion above By a result of Atkinson [I] 7 3 7 implies the Lorenze curve of the after tax income distribu- tion for persons of a given level of training under 7 lies on or above the corresponding curve under 7 However since the investment function e() will depend on 7 the same statement will not generally hold for the overall distribu- tions of income Thus we must use the term more egalitarian with some caution Let (y e) E jh(a e)f(a)p (da) and ~ ( y e) = y for all (y e) E

2+relative to Crepresenting perfect income insurance is maximal in Then on the other hand represents laisset faire For B E (0 I ) let r = B7 + (1 - )I Then r E C Finally for every r E C let V denote the unique solution for (2) when h is replaced by h

THEOREM3 Under A1-A3 7 $ r2 V(y) gt V(y) Vy gt 0 T1T2E C

Theorem 3 states that the use of a more egalitarian PRESTP among future generations of workers will lead to a higher level of welfare for each mature individual currently The careful reader will notice that this result depends upon our assumption that labor supply is exogenous In general we may expect that an income tax will distort both the educational investment decisions of parents and their choices concerning work effort By allowing the income tax function to vary with the level of educational investment (redistributing income within education classes only) we avoid the negative welfare consequences of the former distor- tion It is possible to preserve the result in Theorem 3 in the face of the latter kind of distortion as well if we restrict our attention to small redistributive efforts To see this imagine that h(ae) determines output per unit of time worked that each parent is endowed with one unit of time and that parents utility depends on work effort as well as consumption and offsprings welfare Under these slightly more general circumstances one can repeat the development of Theorems 1 and 2 with the natural modifications that indirect utility becomes a function of output per unit time (instead of total income) and income is the product of work effort and h(a e) Now consider introducing the tax scheme r (described above) for small B For fixed labor supply it may be shown that

854 GLENN C LOURY

welfare at any level of income is a concave function of B Moreover Theorem 3 implies that welfare is an increasing function of B However from standard tax theory we know that the excess burden of the tax T~ is zero when B = 0 and has derivative with respect to equal to zero at B = 0 (That is the excess burden of a small tax is a second order effect) Thus for sufficiently close to zero the insurance effect described in Theorem 3 will outweigh the excess burden effect of the tax scheme leading to an increase in wellbeing for all parents of the current generation

This result suggests that the standard tradeoff between equity and efficiency is somewhat less severe in the second-best world where opportunities to acquire training vary with parental resources It is sometimes possible to design policies which increase both equity and efficiency As a further indication of this last point let us examine the impact of public education in this model By public education we mean the centralized provision of training on an equal basis to every young person in the economy For the purposes of comparison we contrast the laissez faire outcome with that which occurs under public education with a per-capita budget equal to the expenditure of the average family in the no-intervention equilibrium We assume for this result that education is financed with non-distortionary tax

ASSUMPTION is convex in y5 h(ae(y)) is concave in y and ah(ae(y))aa

Assumption 5 posits that the marginal product of social background declines as social background (ie parents income) improves but declines less rapidly for the more able We have proven in Section 6 the following result

THEOREM4 Under A1-A5 universal public education with a per-capita budget equal to the investment of the average family in the laissez faire equilibrium will produce an earnings distribution with lower variance and higher mean than that which obtains without intervention

Finally let us turn to an examination of the relationship between earnings and ability in equilibrium It is widely held that differences in ability provide ethical grounds for differences in rewards An often cited justification for the market determined distribution of economic advantage is that it gives greater rewards to those of greater abilities In our model such is not literally the case because opportunities for training vary with social origin Thus rewards correspond to productivity but not ability Productivity depends both on ability and on social background Nonetheless some sort of systematic positive relationship between earnings and ability might still be sought Imagine that we observe in equilibrium an economy of the type studied here Suppose that we can observe an agents income but neither his ability nor his parents income Now the pair (iux) of individual ability and income is a jointly random vector in the population A very weak definition of a meritocratic distribution of income is that iu and x are positively correlated Under this definition the distribution is meritocratic if a regression of income on ability would yield a positive coefficient in a large

855 INTERGENERATIONAL TRANSFERS

sample A more substantive restriction would require that if x gt x then pr[G gt a 1 x = x ] gt pr[G gt a I x = x2]for all a E [0 11 Under this stronger defi- nition of a meritocratic distribution of rewards we require that greater earnings imply a greater conditional probability that innate aptitude exceeds any pre- specified level The following theorem asserts that the economy studied here is generally weakly meritocratic but not strongly so

THEOREM5 Under A1-A4 the economy is weakly meritocratic However it is not generally true that the economy is strongly meritocratic

The reason that a strong meritocracy may fail to obtain can be easily seen Suppose there exists some range of parental incomes over which the investment schedule e(y) increases rapidly such that the least able offspring of parents at the top earn nearly as much as the most able offspring of parents at the bottom of this range Then the conditional ability distribution of those earning incomes slightly greater than that earned by the most able children of the least well-off parents may be dominated by the conditional distribution of ability of those whose incomes are slightly less than the earnings of the least able children of the most well-off parents An example along these lines is discussed in Section 6

5 AN EXAMPLE

It is interesting to examine the explicit solution of this problem in the special case of Cobb-Douglas production and utility functions For the analysis of this section we assume that

h ( a e ) = (ae)

f ( a ) 3 1 a E [0 1 1

With these particular functions it is not hard to see that the consistent indirect utility function and optimal investment function have the following forms

(10) V ( y ) = k y s k gt 0 0 lt 6 lt 1 and

e (y ) = sy 0 lt s lt 1

The reader may verify by direct substitution that the functions V and e given above satisfy (2)when s = ( 1 + y)2 6 = 2 y ( l + y) and

By Theorem 1 we know these solutions are unique Thus in this simple instance we obtain a constant proportional investment

function This result continues to hold for arbitrary distributions of ability f and

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

845 INTERGENERATIONAL TRANSFERS

a number of economically interesting results are obtained As mentioned risk spreading opportunities permitted by the independence of the random ability endowments across families are explored Moreover inefficiency in the allocation of training resources among the young as a consequence of incomplete loan markets is noted Public provision of training is shown under some conditions to increase output and reduce inequality The section concludes with an examina- tion of the joint distribution of ability and earnings in equilibrium Here we assess the extent to which reward may be presumed to be distributed according to merit We find the notion of meritocracy appropriately defined as difficult to defend when social background constrains individual opportunity In Section 5 we provide a complete solution of our model in the case of Cobb-Douglas utility and production functions and uniform ability distribution The equilibrium distribution is explicitly derived and studied Proofs of the various results appear in Section 6 while concluding observations are mentioned in the final section

2 THE BASIC MODEL

Imagine an economy with overlapping generations along the lines of Samuel- son [26]composed of a large number of individuals Time is measured discretely and each individual lives for exactly two periods An equal number of individu- als enter and leave the economy in each period implying a stationary popula- tion Every person in the first period of life (youth) is attached to a person in the second period of life (maturity) and this union is called a family

The family is the basic socioeconomic unit in this idealized world We take the mature member as family head making all economic decisions The income (output) of a family depends on the productivity of the head which in turn depends upon his training and innate ability6 Family income must be divided in each period between consumption and training of the youth We assume that such investment is the only means of transferring goods through time The training which a mature person possesses this period is simply that portion of family income invested in him last period

Production of the one perishable commodity is a family specific undertaking It requires no social interaction among families nor the use of factors of production other than the mature members time which is supplied inelastically Consumption is also a family activity with young and mature individuals deriving their personal consumption from the family aggregate in a manner determined by social custom and not examined here Natural economic ability differs among individuals each person beginning life with a random endowment of innate aptitudes Moreover the level of this endowment becomes known only in maturity Respectively denoting by x a and e a mature individuals output

6~bility here means all factors outside of the individuals control which affect his productive capacity

We abstract from physical capital and real property so as to focus on intergenerational transmission of earnings capacity

846 GLENN C LOURY

ability and training we write the production possibilities as

(1) x = h ( a e)

The technology h( ) is assumed common to all families in all periods The decision to divide income between consumption and training is taken by

mature people to maximize their expected utility We assume that all agents possess the same utility function The utility of a parent is assumed to depend on family consumption during his tenure and on the wellbeing which the offspring experiences after assuming family leadership9 This wellbeing is taken by parents to be some function of the offsprings future earnings Earnings of offspring for any level of training decided upon by the parent is a random variable determined by (1) and the distribution of a Call the function which parents use to relate the prospective income of their offspring to the childs happiness an indirect utility function Call an indirect utility function consistent if it correctly characterizes the relationship between maximized expected utility and earnings for a parent whenever it is taken to characterize that same relationship for the offspring That is an indirect utility function V is consistent if and only if V solves

[aygt 0 V ( y )= mar E U(c V ( h ( a y - c ) ) ) ] o lt c lt y

where U(c V ) is parental utility when family consumption is c and offspring expected utility is V and E is the expectation operator under the random distribution of innate ability Consistency is assured by (2) because expected utility maximization leads to an indirect utility function for mature individuals identical to that which they presume for their offspring Along with a solution for (2)will come an optimal investment function e(y) which relates parents income to the training of their offspring The properties of these functions are studied below

Notice for now that the earnings of a mature individual ( x ) depends on his random ability and parents earnings ( y )in the following way

Equation (3) is the basic relationship which characterizes social mobility in this economy It establishes a stochastic link between parents and offsprings earn-ings Each individual has limited mobility in that economic achievement varies with aptitude but in a manner determined by parental success Moreover the utility obtained from having a particular level of income V ( y ) depends

he assumption of identical tastes for everyone is purely for convenience It may be replaced with the hypothesis of inheritance of preferences within the family only without affecting any of the formal results

~ l t r u i s m is modelled in this way (ie with the cardinal utility of children entering the parents utility function) rather than with childs consumption in the parents utility function (as in Kohlberg [16]) to avoid the possibly inefficient growth paths associated with the latter set-up (see Diamond [9])

847 INTERGENERATIONAL TRANSFERS

(through equation (2)) on the extent of social mobility This is because inter- generational altruism as modelled here implies that parents are indirectly concerned about the incomes of all of their progeny Thus when there is a great deal of mobility (because a has a large variance andor ahaa is large) a low income will be less of a disadvantage and a high income less of an advantage than when there is very little social mobility Thus the graph of V() will be flatter when the society is more mobile This suggests that our indirect utility function may be useful for assessing the relative merits of social structures generating different degrees of social mobility across generations We shall return to this point below

Summarizing there are three distinct elements which determine the structure of our model economy the utility function the production function and the distribution of the ability endowment We require some assumptions concerning these elements which are informally described here First we shall assume that some consumption is always desirable parents are risk averse and parents care about but discount the wellbeing of their offspring Concerning production we assume diminishing returns to training a strictly positive marginal product of ability that the net marginal return to training eventually becomes negative for all individuals and the net average return is always negative for the least able Finally we assume an atomless distribution of ability with connected and bounded support we rule out autocorrelation in innate endowment^^ and make extreme values of a sufficiently likely that parents will always desire to invest something in their offspring The formal statement of these assumptions follows (R+ is the nonnegative half-line)

ASSUMPTION +R+ is a strictly increasing strictly concave twice 1 (i) U R continuously differentiable function on the interior of its domain satisbing U(0O) = 0 (ii) V V gt 0

lim -au (c V) = +KI ~~0 ac

(iii) There exists y gt 0 such that V(c V) E R (a Ua V) E [y 1 - y ]

ASSUMPTION +R+ is continuously differentiable strictly increasing 2 (i) h R strictly concave in e satisfving h(0O) = 0 and h(0 e) lt e Ve gt 0

ah(ii) - ( a e ) gt p gt O V(ae)euroRaa

OPositive intergenerational correlation of abilities would be a more natural assumption (For a discussion of the evidence see Goldberger [13])We have assumed this away for simplicitys sake One might think this affects the basic character of our argument since if rich parents (and hence their children) are smarter than poor parents it may not be inefficient for them to invest relatively more in their offspring But as our discussion of meritocracy in Section 4 illustrates when parental resources determine human capital investments it is not necessarily true that those with greater income have (probabilistically) greater ability

848 GLENN C LOURY

(iii) There exists p gt 0 I gt 0 such that

ah e gt I max -(a e) 9 p lt 1 a ~ [ o ae11

ASSUMPTION3 Innate economic ability is distributed on the unit interval inde- pendently and identically for all agents The distribution has a continuous strictly positive density function f [0 11 +R+

3 CONSISTENT INDIRECT UTILITY AND THE DYNAMICS OF THE INCOME

DISTRIBUTION

Our assumptions (hereafter Al-A3 etc) enable us to demonstrate the exist- ence of a unique consistent indirect utility function V(y) and to make rather strong statements about the long run behavior of the distribution of earnings implied by equation (3) These essentially mathematical results are required before we can proceed to the economic analysis which is developed in the following section To see that the notion of consistent indirect utility is not vacuous we adopt to our context a dynamic programming argument originated by Bellman First however notice that A2(iii) implies there is a unique earnings level y such that h(1J) = y and y gty h(1 y ) lty Hence we may without loss of generality restrict our attention to incomes in the interval [0 j ] since no familys income could persist outside this range Let C [0 j ] denote the continu- ous real valued functions on [0 y] For 9 E C[O j ] define the function T+() as follows

(4) vy euro [0 y] T+(y) Eo7a E U(c 9(h(a Y - ~ 1 ) ) c y

An indirect utility function is consistent if and only if it is a fixed point of the mapping T Our assumptions permit us to show that T C [0 J] C [0 j ] is a contraction mapping and hence has a unique fixed point In Section 6 we prove the following

THEOREM1 Under Al A2 and A3 for the above defined positive number j there exists a unique consistent indirect utility function V on [0 y] V is strictly increasing strictly concave and differentiable on (0 y] The optimal consumption function c(y) is continuous and satisfies 0 lt c(y) lty Vy E (0 y] (Denote e(y) =y - c(y))

The property of uniqueness established here is very important For neither positive nor normative analysis would our story carry much force were we required to select arbitrarily among several consistent representations of family preferences On the other hand we have assumed a cardinal representation of preferences This is unavoidable if parents are to be taken as contemplating the welfare of their offspring While an axiomatic justification of our choice criterion might be sought along the lines of Koopmans [17] this task is complicated by the

849 INTERGENERATIONAL TRANSFERS

uncertainty inherent in our set-up and is not pursued here As one might expect V inherits the property of risk aversion from U This is important for much of the discussion of Section 4

We turn now to the study of the intergenerational motion of the distribution of income In any period (say nth) the distribution of family income is conceived as a probability measure on [0 j ] v The interpretation is that for A c [0 J] vn(A) is the fraction of all mature agents whose incomes lie in the set A In what follows we denote by 9 the set of probability measures on 9the Bore1 sets of [0 j ] p will denote the Lebesgue measure on 9

The system of family decision making outlined above enables us to treat income dynamics in this economy as a Markoff process To do so we require the following definition

DEFINITION1 A transition probability on [0 j ] is a function P [0 j ] x +[O 11 satisfying (i) Vy E [0 j ] P(y a ) E 9 (ii) VA E P( A) is a -measurable function on R+ To each transition probability on [0 j ] there corresponds a Markoff process That is once a transition probability has been specified one can generate for any fixed v E9a sequence of measures v) as follows

Thus the intergenerational motion of the distribution of income will be fully characterized once we have found the mechanism which relates the probability distribution of an offsprings income to the earnings of its parent Towards this end define the function h - x R++ as follows

Then it is clear that the probability an offspring has income in A when parents earnings is y is simply the probability that the offspring has an innate endow- ment in the set hK1(~e(y)) The relevant transition probability is given by

With P defined as in (6) it is obvious that P(y ) E9moreover it follows from a result of ~ u t i a that P ( A) is -measurable when e() is continuous The last fact was shown in Theorem 1

We now introduce the notion of an equilibrium distribution

DEFINITION2 An equilibrium income distribution is a measure v E 9 satisfy-ing VA E

See Futia [12 Theorem 52 p 741

850 GLENN C LOURY

Thus an equilibrium distribution is one which if it ever characterizes the distribution of earnings among a given generation continues to do so for each subsequent generation Assumptions A1-A3 already permit us to assert the existence of at least one equilibrium and convergence to some equilibrium However since our focus here is on dynamics it seems preferable to introduce the following simple assumption from which follows the global stability of the unique equilibrium point

That is education is a normal good We now have the following theorem

THEOREM2 Under Al-A4 there exists a unique equilibrium distribution v E 9If v) is a sequence of income distributions originating from the arbitrary initial v then limnvn = v pointwise on 9Moreover v has support [OA where y satisfies h (1 e($) =y

Thus by the proof given in Section 6 we establish existence uniqueness and stability for our notion of equilibrium Together with Theorem 1 this constitutes the framework within which we explore the issues raised in the introduction This examination is conducted in the next two sections

4 ECONOMIC ANALYSIS OF THE EQUILIBRIUM DISTRIBUTION

We have now constructed a relatively simple maximizing model of the distribu- tion of lifetime earnings among successive generations of workers The salient features of this model are (i) the recursive family preference structure reflecting intergenerational altruism (ii) the capital-theoretic characterization of produc- tion possibilities (iii) the stochastic nature of innate endowments assumed independent within each family over time and across families at any moment of time and (iv) the assumed absence of markets for human capital loans and income risk sharing between families These factors determine the average level of output in equilibrium and its dispersion as well as the criterion which families use to evaluate their wellbeing

Thus we may think of our model as defining a function M which associates with every triple of utility production and ability distribution functions satisfy- ing our assumptions a unique pair the equilibrium earnings distribution and consistent indirect utility

M (Uh f ) -+(v V)

A natural way to proceed in the analysis of this model is to ask comparative statics questions regarding how the endogenous (v V) vary with changes in the exogenous (U h f) This is a very difficult question in general though we are able to find some preliminary results along these lines presented below

Before proceeding to this however let us notice how the model distinguishes the phenomenon of social mobility across generations from that of inequality in

INTERGENERATIONAL TRANSFERS 851

earnings within a generation The relationship between these alternative notions of social stratification has been the focus of some attention in the literature [4 271 Here the two notions are inextricably intertwined intergenerational social mobil- ity is a property of the transition probability P while cross-sectional inequality is (asymptotically) a property of the equilibrium distribution to which P gives rise The basic problem for a normative analysis of economic inequality raised by this joint determination is that the concept of a more equal income distribution is not easily defined For instance how is one to compare a situation in which there is only a slight degree of inequality among families in any given generation but no mobility within families across generations with a circumstance in which there is substantial intragenerational income dispersion but also a large degree of intergenerational mobility Which situation evidences less inequality In the first case such inequality as exists between families lasts forever while in the latter case family members of a particular cohort experience significant earnings differences but no family is permanently assigned to the bottom or top of the earnings hierarchy

The formulation put forward here provides a way of thinking about this trade-off Our equilibrium notion depicts the intragenerational dispersion in earnings which would persist in the long run On the other hand the consistent indirect utility function incorporates into the valuation which individuals affix to any particular location in the income hierarchy the consequences of subsequent social mobility As noted earlier a great deal of intergenerational mobility implies a relatively flat indirect utility function making the cross-sectional distribution of welfare in equilibrium less unequal than would be the case with little or no mobility Thus by examining the distribution in equilibrium of consistent indirect utility (instead of income) one may simultaneously account for both kinds of inequality within a unified framework Indeed the statistic

is a natural extension of the standard utilitarian criterion and provides a complete ordering of social structures (U h f)satisfying A1-A4

Let us consider now the absence of loan and insurance markets in this model These market imperfections have important allocative consequences A number of writers have called attention to the fact that efficient resource allocation requires that children in low-income families should not be restricted by limited parental resources in their access to training For example Arthur Okun has remarked [22 p 80-811 concerning the modern US economy that The most important consequence (of an imperfect loan market) is the inadequate development of the human resources of the children of poor families-which would judge is one of the most serious inefficiencies of the American Economy today (Emphasis added) While it is certainly the case that in most societies a variety of devices exist for overcoming this problem they are far from perfect For example early childhood investments in nutrition or pre-school education are fundamentally income-constrained Nor should we expect a competitive loan

I

852 GLENN C LOURY

market to completely eliminate the dispersion in expected rates of return to training across families which arises in this model Legally poor parents will not be able to constrain their children to honor debts incurred on their behalf Nor will the newly matured children of wealthy families be able to attach the (human) assets of their less well-off counterparts should the latter decide for whatever reasons not to repay their loans (Default has been a pervasive problem with government guaranteed educational loan programs which would not exist absent public underwriting) Moreover the ability to make use of human capital is unknown even to the borrower Thus genuine bankruptcies can be expected as people mature to learn they are not as able as their parents gambled they would be The absence of inter-family loans in this model reflects an important feature of reality the allocative implications of which deserve study

Another critical feature of this story is that parents bear theoretically insurable risk when investing in their offspring Earnings of children are random here because the economic ability of the child becomes known only in maturity Yet a significant spreading of risk is technologically possible for a group of parents investing at the same level since their childrens earnings are independent identically distributed random variables Again we may expect that markets will fail to bring about complete risk spreading the moral hazard problems of income insurance contracts are o b v i ~ u s ~

Any attempt by a central authority to correct for these market imperfections must cope with the same incentive problems which limit the scope of the competitive mechanism here Efforts by a government to alter the allocation of training among young people will affect the consumption-investment decision of parents Moreover parents may be presumed to have better information than a central authority about the distribution of their childrens abilities (This is especially so if ability is positively correlated across generations of the same family) Yet as long as a group of parents with the same information have different incomes and there is no capital market their investment decisions will vary causing inefficiency This inefficiency could be reduced if the government were to redistribute incomes more equally but only at the cost of the excess burden accompanying such a redistributive tax

However because we have assumed altruism between generations a program of income redistribution which is imposed permanently has welfare effects here not encountered in the usual tax analysis The fact that income will be redistri- buted by taxing the next generation of workers causes each parent to regard his offsprings random income as less risky Since parents are risk averse egalitarian redistributive measures have certain welfare enhancing insurance characteris-tics It may be shown that as long as the redistributive activity is not too extensive this insurance effect dominates the excess burden effect Under such circumstances a permanent redistributive tax policy can be designed in such a way as to make all current mature agents better off

See Zeckhauser [30]for a discussion of these problems

853 INTERGENERATIONAL TRANSFERS

DEFINITION3 An education specific tax policy (ESTP) is a function r R + R+ with the interpretation that r(ye) is the after-tax income of someone whose income is y and training is e An ESTP is admissible if h(a e) = r(h(a e) e) satisfies Assumption 2 An admissible ESTP is purely redistributive (PRESTP) if j[r(h(a e) e) - h ( a e)]f(a)p (da) = 0 for every e gt 0 Denote by C the class of admissible PRESTP C is a non-empty convex set

DEFINITION4 The partial ordering is more egalitarian than denoted + is defined on C by the requirement 7 3 7 if and only if for each e the after tax distribution of income under 7 is more risky in the sense of second order stochastic dominance (see for example [24]) than the after tax distribution under

These two definitions allow us to formalize the discussion above By a result of Atkinson [I] 7 3 7 implies the Lorenze curve of the after tax income distribu- tion for persons of a given level of training under 7 lies on or above the corresponding curve under 7 However since the investment function e() will depend on 7 the same statement will not generally hold for the overall distribu- tions of income Thus we must use the term more egalitarian with some caution Let (y e) E jh(a e)f(a)p (da) and ~ ( y e) = y for all (y e) E

2+relative to Crepresenting perfect income insurance is maximal in Then on the other hand represents laisset faire For B E (0 I ) let r = B7 + (1 - )I Then r E C Finally for every r E C let V denote the unique solution for (2) when h is replaced by h

THEOREM3 Under A1-A3 7 $ r2 V(y) gt V(y) Vy gt 0 T1T2E C

Theorem 3 states that the use of a more egalitarian PRESTP among future generations of workers will lead to a higher level of welfare for each mature individual currently The careful reader will notice that this result depends upon our assumption that labor supply is exogenous In general we may expect that an income tax will distort both the educational investment decisions of parents and their choices concerning work effort By allowing the income tax function to vary with the level of educational investment (redistributing income within education classes only) we avoid the negative welfare consequences of the former distor- tion It is possible to preserve the result in Theorem 3 in the face of the latter kind of distortion as well if we restrict our attention to small redistributive efforts To see this imagine that h(ae) determines output per unit of time worked that each parent is endowed with one unit of time and that parents utility depends on work effort as well as consumption and offsprings welfare Under these slightly more general circumstances one can repeat the development of Theorems 1 and 2 with the natural modifications that indirect utility becomes a function of output per unit time (instead of total income) and income is the product of work effort and h(a e) Now consider introducing the tax scheme r (described above) for small B For fixed labor supply it may be shown that

854 GLENN C LOURY

welfare at any level of income is a concave function of B Moreover Theorem 3 implies that welfare is an increasing function of B However from standard tax theory we know that the excess burden of the tax T~ is zero when B = 0 and has derivative with respect to equal to zero at B = 0 (That is the excess burden of a small tax is a second order effect) Thus for sufficiently close to zero the insurance effect described in Theorem 3 will outweigh the excess burden effect of the tax scheme leading to an increase in wellbeing for all parents of the current generation

This result suggests that the standard tradeoff between equity and efficiency is somewhat less severe in the second-best world where opportunities to acquire training vary with parental resources It is sometimes possible to design policies which increase both equity and efficiency As a further indication of this last point let us examine the impact of public education in this model By public education we mean the centralized provision of training on an equal basis to every young person in the economy For the purposes of comparison we contrast the laissez faire outcome with that which occurs under public education with a per-capita budget equal to the expenditure of the average family in the no-intervention equilibrium We assume for this result that education is financed with non-distortionary tax

ASSUMPTION is convex in y5 h(ae(y)) is concave in y and ah(ae(y))aa

Assumption 5 posits that the marginal product of social background declines as social background (ie parents income) improves but declines less rapidly for the more able We have proven in Section 6 the following result

THEOREM4 Under A1-A5 universal public education with a per-capita budget equal to the investment of the average family in the laissez faire equilibrium will produce an earnings distribution with lower variance and higher mean than that which obtains without intervention

Finally let us turn to an examination of the relationship between earnings and ability in equilibrium It is widely held that differences in ability provide ethical grounds for differences in rewards An often cited justification for the market determined distribution of economic advantage is that it gives greater rewards to those of greater abilities In our model such is not literally the case because opportunities for training vary with social origin Thus rewards correspond to productivity but not ability Productivity depends both on ability and on social background Nonetheless some sort of systematic positive relationship between earnings and ability might still be sought Imagine that we observe in equilibrium an economy of the type studied here Suppose that we can observe an agents income but neither his ability nor his parents income Now the pair (iux) of individual ability and income is a jointly random vector in the population A very weak definition of a meritocratic distribution of income is that iu and x are positively correlated Under this definition the distribution is meritocratic if a regression of income on ability would yield a positive coefficient in a large

855 INTERGENERATIONAL TRANSFERS

sample A more substantive restriction would require that if x gt x then pr[G gt a 1 x = x ] gt pr[G gt a I x = x2]for all a E [0 11 Under this stronger defi- nition of a meritocratic distribution of rewards we require that greater earnings imply a greater conditional probability that innate aptitude exceeds any pre- specified level The following theorem asserts that the economy studied here is generally weakly meritocratic but not strongly so

THEOREM5 Under A1-A4 the economy is weakly meritocratic However it is not generally true that the economy is strongly meritocratic

The reason that a strong meritocracy may fail to obtain can be easily seen Suppose there exists some range of parental incomes over which the investment schedule e(y) increases rapidly such that the least able offspring of parents at the top earn nearly as much as the most able offspring of parents at the bottom of this range Then the conditional ability distribution of those earning incomes slightly greater than that earned by the most able children of the least well-off parents may be dominated by the conditional distribution of ability of those whose incomes are slightly less than the earnings of the least able children of the most well-off parents An example along these lines is discussed in Section 6

5 AN EXAMPLE

It is interesting to examine the explicit solution of this problem in the special case of Cobb-Douglas production and utility functions For the analysis of this section we assume that

h ( a e ) = (ae)

f ( a ) 3 1 a E [0 1 1

With these particular functions it is not hard to see that the consistent indirect utility function and optimal investment function have the following forms

(10) V ( y ) = k y s k gt 0 0 lt 6 lt 1 and

e (y ) = sy 0 lt s lt 1

The reader may verify by direct substitution that the functions V and e given above satisfy (2)when s = ( 1 + y)2 6 = 2 y ( l + y) and

By Theorem 1 we know these solutions are unique Thus in this simple instance we obtain a constant proportional investment

function This result continues to hold for arbitrary distributions of ability f and

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

846 GLENN C LOURY

ability and training we write the production possibilities as

(1) x = h ( a e)

The technology h( ) is assumed common to all families in all periods The decision to divide income between consumption and training is taken by

mature people to maximize their expected utility We assume that all agents possess the same utility function The utility of a parent is assumed to depend on family consumption during his tenure and on the wellbeing which the offspring experiences after assuming family leadership9 This wellbeing is taken by parents to be some function of the offsprings future earnings Earnings of offspring for any level of training decided upon by the parent is a random variable determined by (1) and the distribution of a Call the function which parents use to relate the prospective income of their offspring to the childs happiness an indirect utility function Call an indirect utility function consistent if it correctly characterizes the relationship between maximized expected utility and earnings for a parent whenever it is taken to characterize that same relationship for the offspring That is an indirect utility function V is consistent if and only if V solves

[aygt 0 V ( y )= mar E U(c V ( h ( a y - c ) ) ) ] o lt c lt y

where U(c V ) is parental utility when family consumption is c and offspring expected utility is V and E is the expectation operator under the random distribution of innate ability Consistency is assured by (2) because expected utility maximization leads to an indirect utility function for mature individuals identical to that which they presume for their offspring Along with a solution for (2)will come an optimal investment function e(y) which relates parents income to the training of their offspring The properties of these functions are studied below

Notice for now that the earnings of a mature individual ( x ) depends on his random ability and parents earnings ( y )in the following way

Equation (3) is the basic relationship which characterizes social mobility in this economy It establishes a stochastic link between parents and offsprings earn-ings Each individual has limited mobility in that economic achievement varies with aptitude but in a manner determined by parental success Moreover the utility obtained from having a particular level of income V ( y ) depends

he assumption of identical tastes for everyone is purely for convenience It may be replaced with the hypothesis of inheritance of preferences within the family only without affecting any of the formal results

~ l t r u i s m is modelled in this way (ie with the cardinal utility of children entering the parents utility function) rather than with childs consumption in the parents utility function (as in Kohlberg [16]) to avoid the possibly inefficient growth paths associated with the latter set-up (see Diamond [9])

847 INTERGENERATIONAL TRANSFERS

(through equation (2)) on the extent of social mobility This is because inter- generational altruism as modelled here implies that parents are indirectly concerned about the incomes of all of their progeny Thus when there is a great deal of mobility (because a has a large variance andor ahaa is large) a low income will be less of a disadvantage and a high income less of an advantage than when there is very little social mobility Thus the graph of V() will be flatter when the society is more mobile This suggests that our indirect utility function may be useful for assessing the relative merits of social structures generating different degrees of social mobility across generations We shall return to this point below

Summarizing there are three distinct elements which determine the structure of our model economy the utility function the production function and the distribution of the ability endowment We require some assumptions concerning these elements which are informally described here First we shall assume that some consumption is always desirable parents are risk averse and parents care about but discount the wellbeing of their offspring Concerning production we assume diminishing returns to training a strictly positive marginal product of ability that the net marginal return to training eventually becomes negative for all individuals and the net average return is always negative for the least able Finally we assume an atomless distribution of ability with connected and bounded support we rule out autocorrelation in innate endowment^^ and make extreme values of a sufficiently likely that parents will always desire to invest something in their offspring The formal statement of these assumptions follows (R+ is the nonnegative half-line)

ASSUMPTION +R+ is a strictly increasing strictly concave twice 1 (i) U R continuously differentiable function on the interior of its domain satisbing U(0O) = 0 (ii) V V gt 0

lim -au (c V) = +KI ~~0 ac

(iii) There exists y gt 0 such that V(c V) E R (a Ua V) E [y 1 - y ]

ASSUMPTION +R+ is continuously differentiable strictly increasing 2 (i) h R strictly concave in e satisfving h(0O) = 0 and h(0 e) lt e Ve gt 0

ah(ii) - ( a e ) gt p gt O V(ae)euroRaa

OPositive intergenerational correlation of abilities would be a more natural assumption (For a discussion of the evidence see Goldberger [13])We have assumed this away for simplicitys sake One might think this affects the basic character of our argument since if rich parents (and hence their children) are smarter than poor parents it may not be inefficient for them to invest relatively more in their offspring But as our discussion of meritocracy in Section 4 illustrates when parental resources determine human capital investments it is not necessarily true that those with greater income have (probabilistically) greater ability

848 GLENN C LOURY

(iii) There exists p gt 0 I gt 0 such that

ah e gt I max -(a e) 9 p lt 1 a ~ [ o ae11

ASSUMPTION3 Innate economic ability is distributed on the unit interval inde- pendently and identically for all agents The distribution has a continuous strictly positive density function f [0 11 +R+

3 CONSISTENT INDIRECT UTILITY AND THE DYNAMICS OF THE INCOME

DISTRIBUTION

Our assumptions (hereafter Al-A3 etc) enable us to demonstrate the exist- ence of a unique consistent indirect utility function V(y) and to make rather strong statements about the long run behavior of the distribution of earnings implied by equation (3) These essentially mathematical results are required before we can proceed to the economic analysis which is developed in the following section To see that the notion of consistent indirect utility is not vacuous we adopt to our context a dynamic programming argument originated by Bellman First however notice that A2(iii) implies there is a unique earnings level y such that h(1J) = y and y gty h(1 y ) lty Hence we may without loss of generality restrict our attention to incomes in the interval [0 j ] since no familys income could persist outside this range Let C [0 j ] denote the continu- ous real valued functions on [0 y] For 9 E C[O j ] define the function T+() as follows

(4) vy euro [0 y] T+(y) Eo7a E U(c 9(h(a Y - ~ 1 ) ) c y

An indirect utility function is consistent if and only if it is a fixed point of the mapping T Our assumptions permit us to show that T C [0 J] C [0 j ] is a contraction mapping and hence has a unique fixed point In Section 6 we prove the following

THEOREM1 Under Al A2 and A3 for the above defined positive number j there exists a unique consistent indirect utility function V on [0 y] V is strictly increasing strictly concave and differentiable on (0 y] The optimal consumption function c(y) is continuous and satisfies 0 lt c(y) lty Vy E (0 y] (Denote e(y) =y - c(y))

The property of uniqueness established here is very important For neither positive nor normative analysis would our story carry much force were we required to select arbitrarily among several consistent representations of family preferences On the other hand we have assumed a cardinal representation of preferences This is unavoidable if parents are to be taken as contemplating the welfare of their offspring While an axiomatic justification of our choice criterion might be sought along the lines of Koopmans [17] this task is complicated by the

849 INTERGENERATIONAL TRANSFERS

uncertainty inherent in our set-up and is not pursued here As one might expect V inherits the property of risk aversion from U This is important for much of the discussion of Section 4

We turn now to the study of the intergenerational motion of the distribution of income In any period (say nth) the distribution of family income is conceived as a probability measure on [0 j ] v The interpretation is that for A c [0 J] vn(A) is the fraction of all mature agents whose incomes lie in the set A In what follows we denote by 9 the set of probability measures on 9the Bore1 sets of [0 j ] p will denote the Lebesgue measure on 9

The system of family decision making outlined above enables us to treat income dynamics in this economy as a Markoff process To do so we require the following definition

DEFINITION1 A transition probability on [0 j ] is a function P [0 j ] x +[O 11 satisfying (i) Vy E [0 j ] P(y a ) E 9 (ii) VA E P( A) is a -measurable function on R+ To each transition probability on [0 j ] there corresponds a Markoff process That is once a transition probability has been specified one can generate for any fixed v E9a sequence of measures v) as follows

Thus the intergenerational motion of the distribution of income will be fully characterized once we have found the mechanism which relates the probability distribution of an offsprings income to the earnings of its parent Towards this end define the function h - x R++ as follows

Then it is clear that the probability an offspring has income in A when parents earnings is y is simply the probability that the offspring has an innate endow- ment in the set hK1(~e(y)) The relevant transition probability is given by

With P defined as in (6) it is obvious that P(y ) E9moreover it follows from a result of ~ u t i a that P ( A) is -measurable when e() is continuous The last fact was shown in Theorem 1

We now introduce the notion of an equilibrium distribution

DEFINITION2 An equilibrium income distribution is a measure v E 9 satisfy-ing VA E

See Futia [12 Theorem 52 p 741

850 GLENN C LOURY

Thus an equilibrium distribution is one which if it ever characterizes the distribution of earnings among a given generation continues to do so for each subsequent generation Assumptions A1-A3 already permit us to assert the existence of at least one equilibrium and convergence to some equilibrium However since our focus here is on dynamics it seems preferable to introduce the following simple assumption from which follows the global stability of the unique equilibrium point

That is education is a normal good We now have the following theorem

THEOREM2 Under Al-A4 there exists a unique equilibrium distribution v E 9If v) is a sequence of income distributions originating from the arbitrary initial v then limnvn = v pointwise on 9Moreover v has support [OA where y satisfies h (1 e($) =y

Thus by the proof given in Section 6 we establish existence uniqueness and stability for our notion of equilibrium Together with Theorem 1 this constitutes the framework within which we explore the issues raised in the introduction This examination is conducted in the next two sections

4 ECONOMIC ANALYSIS OF THE EQUILIBRIUM DISTRIBUTION

We have now constructed a relatively simple maximizing model of the distribu- tion of lifetime earnings among successive generations of workers The salient features of this model are (i) the recursive family preference structure reflecting intergenerational altruism (ii) the capital-theoretic characterization of produc- tion possibilities (iii) the stochastic nature of innate endowments assumed independent within each family over time and across families at any moment of time and (iv) the assumed absence of markets for human capital loans and income risk sharing between families These factors determine the average level of output in equilibrium and its dispersion as well as the criterion which families use to evaluate their wellbeing

Thus we may think of our model as defining a function M which associates with every triple of utility production and ability distribution functions satisfy- ing our assumptions a unique pair the equilibrium earnings distribution and consistent indirect utility

M (Uh f ) -+(v V)

A natural way to proceed in the analysis of this model is to ask comparative statics questions regarding how the endogenous (v V) vary with changes in the exogenous (U h f) This is a very difficult question in general though we are able to find some preliminary results along these lines presented below

Before proceeding to this however let us notice how the model distinguishes the phenomenon of social mobility across generations from that of inequality in

INTERGENERATIONAL TRANSFERS 851

earnings within a generation The relationship between these alternative notions of social stratification has been the focus of some attention in the literature [4 271 Here the two notions are inextricably intertwined intergenerational social mobil- ity is a property of the transition probability P while cross-sectional inequality is (asymptotically) a property of the equilibrium distribution to which P gives rise The basic problem for a normative analysis of economic inequality raised by this joint determination is that the concept of a more equal income distribution is not easily defined For instance how is one to compare a situation in which there is only a slight degree of inequality among families in any given generation but no mobility within families across generations with a circumstance in which there is substantial intragenerational income dispersion but also a large degree of intergenerational mobility Which situation evidences less inequality In the first case such inequality as exists between families lasts forever while in the latter case family members of a particular cohort experience significant earnings differences but no family is permanently assigned to the bottom or top of the earnings hierarchy

The formulation put forward here provides a way of thinking about this trade-off Our equilibrium notion depicts the intragenerational dispersion in earnings which would persist in the long run On the other hand the consistent indirect utility function incorporates into the valuation which individuals affix to any particular location in the income hierarchy the consequences of subsequent social mobility As noted earlier a great deal of intergenerational mobility implies a relatively flat indirect utility function making the cross-sectional distribution of welfare in equilibrium less unequal than would be the case with little or no mobility Thus by examining the distribution in equilibrium of consistent indirect utility (instead of income) one may simultaneously account for both kinds of inequality within a unified framework Indeed the statistic

is a natural extension of the standard utilitarian criterion and provides a complete ordering of social structures (U h f)satisfying A1-A4

Let us consider now the absence of loan and insurance markets in this model These market imperfections have important allocative consequences A number of writers have called attention to the fact that efficient resource allocation requires that children in low-income families should not be restricted by limited parental resources in their access to training For example Arthur Okun has remarked [22 p 80-811 concerning the modern US economy that The most important consequence (of an imperfect loan market) is the inadequate development of the human resources of the children of poor families-which would judge is one of the most serious inefficiencies of the American Economy today (Emphasis added) While it is certainly the case that in most societies a variety of devices exist for overcoming this problem they are far from perfect For example early childhood investments in nutrition or pre-school education are fundamentally income-constrained Nor should we expect a competitive loan

I

852 GLENN C LOURY

market to completely eliminate the dispersion in expected rates of return to training across families which arises in this model Legally poor parents will not be able to constrain their children to honor debts incurred on their behalf Nor will the newly matured children of wealthy families be able to attach the (human) assets of their less well-off counterparts should the latter decide for whatever reasons not to repay their loans (Default has been a pervasive problem with government guaranteed educational loan programs which would not exist absent public underwriting) Moreover the ability to make use of human capital is unknown even to the borrower Thus genuine bankruptcies can be expected as people mature to learn they are not as able as their parents gambled they would be The absence of inter-family loans in this model reflects an important feature of reality the allocative implications of which deserve study

Another critical feature of this story is that parents bear theoretically insurable risk when investing in their offspring Earnings of children are random here because the economic ability of the child becomes known only in maturity Yet a significant spreading of risk is technologically possible for a group of parents investing at the same level since their childrens earnings are independent identically distributed random variables Again we may expect that markets will fail to bring about complete risk spreading the moral hazard problems of income insurance contracts are o b v i ~ u s ~

Any attempt by a central authority to correct for these market imperfections must cope with the same incentive problems which limit the scope of the competitive mechanism here Efforts by a government to alter the allocation of training among young people will affect the consumption-investment decision of parents Moreover parents may be presumed to have better information than a central authority about the distribution of their childrens abilities (This is especially so if ability is positively correlated across generations of the same family) Yet as long as a group of parents with the same information have different incomes and there is no capital market their investment decisions will vary causing inefficiency This inefficiency could be reduced if the government were to redistribute incomes more equally but only at the cost of the excess burden accompanying such a redistributive tax

However because we have assumed altruism between generations a program of income redistribution which is imposed permanently has welfare effects here not encountered in the usual tax analysis The fact that income will be redistri- buted by taxing the next generation of workers causes each parent to regard his offsprings random income as less risky Since parents are risk averse egalitarian redistributive measures have certain welfare enhancing insurance characteris-tics It may be shown that as long as the redistributive activity is not too extensive this insurance effect dominates the excess burden effect Under such circumstances a permanent redistributive tax policy can be designed in such a way as to make all current mature agents better off

See Zeckhauser [30]for a discussion of these problems

853 INTERGENERATIONAL TRANSFERS

DEFINITION3 An education specific tax policy (ESTP) is a function r R + R+ with the interpretation that r(ye) is the after-tax income of someone whose income is y and training is e An ESTP is admissible if h(a e) = r(h(a e) e) satisfies Assumption 2 An admissible ESTP is purely redistributive (PRESTP) if j[r(h(a e) e) - h ( a e)]f(a)p (da) = 0 for every e gt 0 Denote by C the class of admissible PRESTP C is a non-empty convex set

DEFINITION4 The partial ordering is more egalitarian than denoted + is defined on C by the requirement 7 3 7 if and only if for each e the after tax distribution of income under 7 is more risky in the sense of second order stochastic dominance (see for example [24]) than the after tax distribution under

These two definitions allow us to formalize the discussion above By a result of Atkinson [I] 7 3 7 implies the Lorenze curve of the after tax income distribu- tion for persons of a given level of training under 7 lies on or above the corresponding curve under 7 However since the investment function e() will depend on 7 the same statement will not generally hold for the overall distribu- tions of income Thus we must use the term more egalitarian with some caution Let (y e) E jh(a e)f(a)p (da) and ~ ( y e) = y for all (y e) E

2+relative to Crepresenting perfect income insurance is maximal in Then on the other hand represents laisset faire For B E (0 I ) let r = B7 + (1 - )I Then r E C Finally for every r E C let V denote the unique solution for (2) when h is replaced by h

THEOREM3 Under A1-A3 7 $ r2 V(y) gt V(y) Vy gt 0 T1T2E C

Theorem 3 states that the use of a more egalitarian PRESTP among future generations of workers will lead to a higher level of welfare for each mature individual currently The careful reader will notice that this result depends upon our assumption that labor supply is exogenous In general we may expect that an income tax will distort both the educational investment decisions of parents and their choices concerning work effort By allowing the income tax function to vary with the level of educational investment (redistributing income within education classes only) we avoid the negative welfare consequences of the former distor- tion It is possible to preserve the result in Theorem 3 in the face of the latter kind of distortion as well if we restrict our attention to small redistributive efforts To see this imagine that h(ae) determines output per unit of time worked that each parent is endowed with one unit of time and that parents utility depends on work effort as well as consumption and offsprings welfare Under these slightly more general circumstances one can repeat the development of Theorems 1 and 2 with the natural modifications that indirect utility becomes a function of output per unit time (instead of total income) and income is the product of work effort and h(a e) Now consider introducing the tax scheme r (described above) for small B For fixed labor supply it may be shown that

854 GLENN C LOURY

welfare at any level of income is a concave function of B Moreover Theorem 3 implies that welfare is an increasing function of B However from standard tax theory we know that the excess burden of the tax T~ is zero when B = 0 and has derivative with respect to equal to zero at B = 0 (That is the excess burden of a small tax is a second order effect) Thus for sufficiently close to zero the insurance effect described in Theorem 3 will outweigh the excess burden effect of the tax scheme leading to an increase in wellbeing for all parents of the current generation

This result suggests that the standard tradeoff between equity and efficiency is somewhat less severe in the second-best world where opportunities to acquire training vary with parental resources It is sometimes possible to design policies which increase both equity and efficiency As a further indication of this last point let us examine the impact of public education in this model By public education we mean the centralized provision of training on an equal basis to every young person in the economy For the purposes of comparison we contrast the laissez faire outcome with that which occurs under public education with a per-capita budget equal to the expenditure of the average family in the no-intervention equilibrium We assume for this result that education is financed with non-distortionary tax

ASSUMPTION is convex in y5 h(ae(y)) is concave in y and ah(ae(y))aa

Assumption 5 posits that the marginal product of social background declines as social background (ie parents income) improves but declines less rapidly for the more able We have proven in Section 6 the following result

THEOREM4 Under A1-A5 universal public education with a per-capita budget equal to the investment of the average family in the laissez faire equilibrium will produce an earnings distribution with lower variance and higher mean than that which obtains without intervention

Finally let us turn to an examination of the relationship between earnings and ability in equilibrium It is widely held that differences in ability provide ethical grounds for differences in rewards An often cited justification for the market determined distribution of economic advantage is that it gives greater rewards to those of greater abilities In our model such is not literally the case because opportunities for training vary with social origin Thus rewards correspond to productivity but not ability Productivity depends both on ability and on social background Nonetheless some sort of systematic positive relationship between earnings and ability might still be sought Imagine that we observe in equilibrium an economy of the type studied here Suppose that we can observe an agents income but neither his ability nor his parents income Now the pair (iux) of individual ability and income is a jointly random vector in the population A very weak definition of a meritocratic distribution of income is that iu and x are positively correlated Under this definition the distribution is meritocratic if a regression of income on ability would yield a positive coefficient in a large

855 INTERGENERATIONAL TRANSFERS

sample A more substantive restriction would require that if x gt x then pr[G gt a 1 x = x ] gt pr[G gt a I x = x2]for all a E [0 11 Under this stronger defi- nition of a meritocratic distribution of rewards we require that greater earnings imply a greater conditional probability that innate aptitude exceeds any pre- specified level The following theorem asserts that the economy studied here is generally weakly meritocratic but not strongly so

THEOREM5 Under A1-A4 the economy is weakly meritocratic However it is not generally true that the economy is strongly meritocratic

The reason that a strong meritocracy may fail to obtain can be easily seen Suppose there exists some range of parental incomes over which the investment schedule e(y) increases rapidly such that the least able offspring of parents at the top earn nearly as much as the most able offspring of parents at the bottom of this range Then the conditional ability distribution of those earning incomes slightly greater than that earned by the most able children of the least well-off parents may be dominated by the conditional distribution of ability of those whose incomes are slightly less than the earnings of the least able children of the most well-off parents An example along these lines is discussed in Section 6

5 AN EXAMPLE

It is interesting to examine the explicit solution of this problem in the special case of Cobb-Douglas production and utility functions For the analysis of this section we assume that

h ( a e ) = (ae)

f ( a ) 3 1 a E [0 1 1

With these particular functions it is not hard to see that the consistent indirect utility function and optimal investment function have the following forms

(10) V ( y ) = k y s k gt 0 0 lt 6 lt 1 and

e (y ) = sy 0 lt s lt 1

The reader may verify by direct substitution that the functions V and e given above satisfy (2)when s = ( 1 + y)2 6 = 2 y ( l + y) and

By Theorem 1 we know these solutions are unique Thus in this simple instance we obtain a constant proportional investment

function This result continues to hold for arbitrary distributions of ability f and

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

847 INTERGENERATIONAL TRANSFERS

(through equation (2)) on the extent of social mobility This is because inter- generational altruism as modelled here implies that parents are indirectly concerned about the incomes of all of their progeny Thus when there is a great deal of mobility (because a has a large variance andor ahaa is large) a low income will be less of a disadvantage and a high income less of an advantage than when there is very little social mobility Thus the graph of V() will be flatter when the society is more mobile This suggests that our indirect utility function may be useful for assessing the relative merits of social structures generating different degrees of social mobility across generations We shall return to this point below

Summarizing there are three distinct elements which determine the structure of our model economy the utility function the production function and the distribution of the ability endowment We require some assumptions concerning these elements which are informally described here First we shall assume that some consumption is always desirable parents are risk averse and parents care about but discount the wellbeing of their offspring Concerning production we assume diminishing returns to training a strictly positive marginal product of ability that the net marginal return to training eventually becomes negative for all individuals and the net average return is always negative for the least able Finally we assume an atomless distribution of ability with connected and bounded support we rule out autocorrelation in innate endowment^^ and make extreme values of a sufficiently likely that parents will always desire to invest something in their offspring The formal statement of these assumptions follows (R+ is the nonnegative half-line)

ASSUMPTION +R+ is a strictly increasing strictly concave twice 1 (i) U R continuously differentiable function on the interior of its domain satisbing U(0O) = 0 (ii) V V gt 0

lim -au (c V) = +KI ~~0 ac

(iii) There exists y gt 0 such that V(c V) E R (a Ua V) E [y 1 - y ]

ASSUMPTION +R+ is continuously differentiable strictly increasing 2 (i) h R strictly concave in e satisfving h(0O) = 0 and h(0 e) lt e Ve gt 0

ah(ii) - ( a e ) gt p gt O V(ae)euroRaa

OPositive intergenerational correlation of abilities would be a more natural assumption (For a discussion of the evidence see Goldberger [13])We have assumed this away for simplicitys sake One might think this affects the basic character of our argument since if rich parents (and hence their children) are smarter than poor parents it may not be inefficient for them to invest relatively more in their offspring But as our discussion of meritocracy in Section 4 illustrates when parental resources determine human capital investments it is not necessarily true that those with greater income have (probabilistically) greater ability

848 GLENN C LOURY

(iii) There exists p gt 0 I gt 0 such that

ah e gt I max -(a e) 9 p lt 1 a ~ [ o ae11

ASSUMPTION3 Innate economic ability is distributed on the unit interval inde- pendently and identically for all agents The distribution has a continuous strictly positive density function f [0 11 +R+

3 CONSISTENT INDIRECT UTILITY AND THE DYNAMICS OF THE INCOME

DISTRIBUTION

Our assumptions (hereafter Al-A3 etc) enable us to demonstrate the exist- ence of a unique consistent indirect utility function V(y) and to make rather strong statements about the long run behavior of the distribution of earnings implied by equation (3) These essentially mathematical results are required before we can proceed to the economic analysis which is developed in the following section To see that the notion of consistent indirect utility is not vacuous we adopt to our context a dynamic programming argument originated by Bellman First however notice that A2(iii) implies there is a unique earnings level y such that h(1J) = y and y gty h(1 y ) lty Hence we may without loss of generality restrict our attention to incomes in the interval [0 j ] since no familys income could persist outside this range Let C [0 j ] denote the continu- ous real valued functions on [0 y] For 9 E C[O j ] define the function T+() as follows

(4) vy euro [0 y] T+(y) Eo7a E U(c 9(h(a Y - ~ 1 ) ) c y

An indirect utility function is consistent if and only if it is a fixed point of the mapping T Our assumptions permit us to show that T C [0 J] C [0 j ] is a contraction mapping and hence has a unique fixed point In Section 6 we prove the following

THEOREM1 Under Al A2 and A3 for the above defined positive number j there exists a unique consistent indirect utility function V on [0 y] V is strictly increasing strictly concave and differentiable on (0 y] The optimal consumption function c(y) is continuous and satisfies 0 lt c(y) lty Vy E (0 y] (Denote e(y) =y - c(y))

The property of uniqueness established here is very important For neither positive nor normative analysis would our story carry much force were we required to select arbitrarily among several consistent representations of family preferences On the other hand we have assumed a cardinal representation of preferences This is unavoidable if parents are to be taken as contemplating the welfare of their offspring While an axiomatic justification of our choice criterion might be sought along the lines of Koopmans [17] this task is complicated by the

849 INTERGENERATIONAL TRANSFERS

uncertainty inherent in our set-up and is not pursued here As one might expect V inherits the property of risk aversion from U This is important for much of the discussion of Section 4

We turn now to the study of the intergenerational motion of the distribution of income In any period (say nth) the distribution of family income is conceived as a probability measure on [0 j ] v The interpretation is that for A c [0 J] vn(A) is the fraction of all mature agents whose incomes lie in the set A In what follows we denote by 9 the set of probability measures on 9the Bore1 sets of [0 j ] p will denote the Lebesgue measure on 9

The system of family decision making outlined above enables us to treat income dynamics in this economy as a Markoff process To do so we require the following definition

DEFINITION1 A transition probability on [0 j ] is a function P [0 j ] x +[O 11 satisfying (i) Vy E [0 j ] P(y a ) E 9 (ii) VA E P( A) is a -measurable function on R+ To each transition probability on [0 j ] there corresponds a Markoff process That is once a transition probability has been specified one can generate for any fixed v E9a sequence of measures v) as follows

Thus the intergenerational motion of the distribution of income will be fully characterized once we have found the mechanism which relates the probability distribution of an offsprings income to the earnings of its parent Towards this end define the function h - x R++ as follows

Then it is clear that the probability an offspring has income in A when parents earnings is y is simply the probability that the offspring has an innate endow- ment in the set hK1(~e(y)) The relevant transition probability is given by

With P defined as in (6) it is obvious that P(y ) E9moreover it follows from a result of ~ u t i a that P ( A) is -measurable when e() is continuous The last fact was shown in Theorem 1

We now introduce the notion of an equilibrium distribution

DEFINITION2 An equilibrium income distribution is a measure v E 9 satisfy-ing VA E

See Futia [12 Theorem 52 p 741

850 GLENN C LOURY

Thus an equilibrium distribution is one which if it ever characterizes the distribution of earnings among a given generation continues to do so for each subsequent generation Assumptions A1-A3 already permit us to assert the existence of at least one equilibrium and convergence to some equilibrium However since our focus here is on dynamics it seems preferable to introduce the following simple assumption from which follows the global stability of the unique equilibrium point

That is education is a normal good We now have the following theorem

THEOREM2 Under Al-A4 there exists a unique equilibrium distribution v E 9If v) is a sequence of income distributions originating from the arbitrary initial v then limnvn = v pointwise on 9Moreover v has support [OA where y satisfies h (1 e($) =y

Thus by the proof given in Section 6 we establish existence uniqueness and stability for our notion of equilibrium Together with Theorem 1 this constitutes the framework within which we explore the issues raised in the introduction This examination is conducted in the next two sections

4 ECONOMIC ANALYSIS OF THE EQUILIBRIUM DISTRIBUTION

We have now constructed a relatively simple maximizing model of the distribu- tion of lifetime earnings among successive generations of workers The salient features of this model are (i) the recursive family preference structure reflecting intergenerational altruism (ii) the capital-theoretic characterization of produc- tion possibilities (iii) the stochastic nature of innate endowments assumed independent within each family over time and across families at any moment of time and (iv) the assumed absence of markets for human capital loans and income risk sharing between families These factors determine the average level of output in equilibrium and its dispersion as well as the criterion which families use to evaluate their wellbeing

Thus we may think of our model as defining a function M which associates with every triple of utility production and ability distribution functions satisfy- ing our assumptions a unique pair the equilibrium earnings distribution and consistent indirect utility

M (Uh f ) -+(v V)

A natural way to proceed in the analysis of this model is to ask comparative statics questions regarding how the endogenous (v V) vary with changes in the exogenous (U h f) This is a very difficult question in general though we are able to find some preliminary results along these lines presented below

Before proceeding to this however let us notice how the model distinguishes the phenomenon of social mobility across generations from that of inequality in

INTERGENERATIONAL TRANSFERS 851

earnings within a generation The relationship between these alternative notions of social stratification has been the focus of some attention in the literature [4 271 Here the two notions are inextricably intertwined intergenerational social mobil- ity is a property of the transition probability P while cross-sectional inequality is (asymptotically) a property of the equilibrium distribution to which P gives rise The basic problem for a normative analysis of economic inequality raised by this joint determination is that the concept of a more equal income distribution is not easily defined For instance how is one to compare a situation in which there is only a slight degree of inequality among families in any given generation but no mobility within families across generations with a circumstance in which there is substantial intragenerational income dispersion but also a large degree of intergenerational mobility Which situation evidences less inequality In the first case such inequality as exists between families lasts forever while in the latter case family members of a particular cohort experience significant earnings differences but no family is permanently assigned to the bottom or top of the earnings hierarchy

The formulation put forward here provides a way of thinking about this trade-off Our equilibrium notion depicts the intragenerational dispersion in earnings which would persist in the long run On the other hand the consistent indirect utility function incorporates into the valuation which individuals affix to any particular location in the income hierarchy the consequences of subsequent social mobility As noted earlier a great deal of intergenerational mobility implies a relatively flat indirect utility function making the cross-sectional distribution of welfare in equilibrium less unequal than would be the case with little or no mobility Thus by examining the distribution in equilibrium of consistent indirect utility (instead of income) one may simultaneously account for both kinds of inequality within a unified framework Indeed the statistic

is a natural extension of the standard utilitarian criterion and provides a complete ordering of social structures (U h f)satisfying A1-A4

Let us consider now the absence of loan and insurance markets in this model These market imperfections have important allocative consequences A number of writers have called attention to the fact that efficient resource allocation requires that children in low-income families should not be restricted by limited parental resources in their access to training For example Arthur Okun has remarked [22 p 80-811 concerning the modern US economy that The most important consequence (of an imperfect loan market) is the inadequate development of the human resources of the children of poor families-which would judge is one of the most serious inefficiencies of the American Economy today (Emphasis added) While it is certainly the case that in most societies a variety of devices exist for overcoming this problem they are far from perfect For example early childhood investments in nutrition or pre-school education are fundamentally income-constrained Nor should we expect a competitive loan

I

852 GLENN C LOURY

market to completely eliminate the dispersion in expected rates of return to training across families which arises in this model Legally poor parents will not be able to constrain their children to honor debts incurred on their behalf Nor will the newly matured children of wealthy families be able to attach the (human) assets of their less well-off counterparts should the latter decide for whatever reasons not to repay their loans (Default has been a pervasive problem with government guaranteed educational loan programs which would not exist absent public underwriting) Moreover the ability to make use of human capital is unknown even to the borrower Thus genuine bankruptcies can be expected as people mature to learn they are not as able as their parents gambled they would be The absence of inter-family loans in this model reflects an important feature of reality the allocative implications of which deserve study

Another critical feature of this story is that parents bear theoretically insurable risk when investing in their offspring Earnings of children are random here because the economic ability of the child becomes known only in maturity Yet a significant spreading of risk is technologically possible for a group of parents investing at the same level since their childrens earnings are independent identically distributed random variables Again we may expect that markets will fail to bring about complete risk spreading the moral hazard problems of income insurance contracts are o b v i ~ u s ~

Any attempt by a central authority to correct for these market imperfections must cope with the same incentive problems which limit the scope of the competitive mechanism here Efforts by a government to alter the allocation of training among young people will affect the consumption-investment decision of parents Moreover parents may be presumed to have better information than a central authority about the distribution of their childrens abilities (This is especially so if ability is positively correlated across generations of the same family) Yet as long as a group of parents with the same information have different incomes and there is no capital market their investment decisions will vary causing inefficiency This inefficiency could be reduced if the government were to redistribute incomes more equally but only at the cost of the excess burden accompanying such a redistributive tax

However because we have assumed altruism between generations a program of income redistribution which is imposed permanently has welfare effects here not encountered in the usual tax analysis The fact that income will be redistri- buted by taxing the next generation of workers causes each parent to regard his offsprings random income as less risky Since parents are risk averse egalitarian redistributive measures have certain welfare enhancing insurance characteris-tics It may be shown that as long as the redistributive activity is not too extensive this insurance effect dominates the excess burden effect Under such circumstances a permanent redistributive tax policy can be designed in such a way as to make all current mature agents better off

See Zeckhauser [30]for a discussion of these problems

853 INTERGENERATIONAL TRANSFERS

DEFINITION3 An education specific tax policy (ESTP) is a function r R + R+ with the interpretation that r(ye) is the after-tax income of someone whose income is y and training is e An ESTP is admissible if h(a e) = r(h(a e) e) satisfies Assumption 2 An admissible ESTP is purely redistributive (PRESTP) if j[r(h(a e) e) - h ( a e)]f(a)p (da) = 0 for every e gt 0 Denote by C the class of admissible PRESTP C is a non-empty convex set

DEFINITION4 The partial ordering is more egalitarian than denoted + is defined on C by the requirement 7 3 7 if and only if for each e the after tax distribution of income under 7 is more risky in the sense of second order stochastic dominance (see for example [24]) than the after tax distribution under

These two definitions allow us to formalize the discussion above By a result of Atkinson [I] 7 3 7 implies the Lorenze curve of the after tax income distribu- tion for persons of a given level of training under 7 lies on or above the corresponding curve under 7 However since the investment function e() will depend on 7 the same statement will not generally hold for the overall distribu- tions of income Thus we must use the term more egalitarian with some caution Let (y e) E jh(a e)f(a)p (da) and ~ ( y e) = y for all (y e) E

2+relative to Crepresenting perfect income insurance is maximal in Then on the other hand represents laisset faire For B E (0 I ) let r = B7 + (1 - )I Then r E C Finally for every r E C let V denote the unique solution for (2) when h is replaced by h

THEOREM3 Under A1-A3 7 $ r2 V(y) gt V(y) Vy gt 0 T1T2E C

Theorem 3 states that the use of a more egalitarian PRESTP among future generations of workers will lead to a higher level of welfare for each mature individual currently The careful reader will notice that this result depends upon our assumption that labor supply is exogenous In general we may expect that an income tax will distort both the educational investment decisions of parents and their choices concerning work effort By allowing the income tax function to vary with the level of educational investment (redistributing income within education classes only) we avoid the negative welfare consequences of the former distor- tion It is possible to preserve the result in Theorem 3 in the face of the latter kind of distortion as well if we restrict our attention to small redistributive efforts To see this imagine that h(ae) determines output per unit of time worked that each parent is endowed with one unit of time and that parents utility depends on work effort as well as consumption and offsprings welfare Under these slightly more general circumstances one can repeat the development of Theorems 1 and 2 with the natural modifications that indirect utility becomes a function of output per unit time (instead of total income) and income is the product of work effort and h(a e) Now consider introducing the tax scheme r (described above) for small B For fixed labor supply it may be shown that

854 GLENN C LOURY

welfare at any level of income is a concave function of B Moreover Theorem 3 implies that welfare is an increasing function of B However from standard tax theory we know that the excess burden of the tax T~ is zero when B = 0 and has derivative with respect to equal to zero at B = 0 (That is the excess burden of a small tax is a second order effect) Thus for sufficiently close to zero the insurance effect described in Theorem 3 will outweigh the excess burden effect of the tax scheme leading to an increase in wellbeing for all parents of the current generation

This result suggests that the standard tradeoff between equity and efficiency is somewhat less severe in the second-best world where opportunities to acquire training vary with parental resources It is sometimes possible to design policies which increase both equity and efficiency As a further indication of this last point let us examine the impact of public education in this model By public education we mean the centralized provision of training on an equal basis to every young person in the economy For the purposes of comparison we contrast the laissez faire outcome with that which occurs under public education with a per-capita budget equal to the expenditure of the average family in the no-intervention equilibrium We assume for this result that education is financed with non-distortionary tax

ASSUMPTION is convex in y5 h(ae(y)) is concave in y and ah(ae(y))aa

Assumption 5 posits that the marginal product of social background declines as social background (ie parents income) improves but declines less rapidly for the more able We have proven in Section 6 the following result

THEOREM4 Under A1-A5 universal public education with a per-capita budget equal to the investment of the average family in the laissez faire equilibrium will produce an earnings distribution with lower variance and higher mean than that which obtains without intervention

Finally let us turn to an examination of the relationship between earnings and ability in equilibrium It is widely held that differences in ability provide ethical grounds for differences in rewards An often cited justification for the market determined distribution of economic advantage is that it gives greater rewards to those of greater abilities In our model such is not literally the case because opportunities for training vary with social origin Thus rewards correspond to productivity but not ability Productivity depends both on ability and on social background Nonetheless some sort of systematic positive relationship between earnings and ability might still be sought Imagine that we observe in equilibrium an economy of the type studied here Suppose that we can observe an agents income but neither his ability nor his parents income Now the pair (iux) of individual ability and income is a jointly random vector in the population A very weak definition of a meritocratic distribution of income is that iu and x are positively correlated Under this definition the distribution is meritocratic if a regression of income on ability would yield a positive coefficient in a large

855 INTERGENERATIONAL TRANSFERS

sample A more substantive restriction would require that if x gt x then pr[G gt a 1 x = x ] gt pr[G gt a I x = x2]for all a E [0 11 Under this stronger defi- nition of a meritocratic distribution of rewards we require that greater earnings imply a greater conditional probability that innate aptitude exceeds any pre- specified level The following theorem asserts that the economy studied here is generally weakly meritocratic but not strongly so

THEOREM5 Under A1-A4 the economy is weakly meritocratic However it is not generally true that the economy is strongly meritocratic

The reason that a strong meritocracy may fail to obtain can be easily seen Suppose there exists some range of parental incomes over which the investment schedule e(y) increases rapidly such that the least able offspring of parents at the top earn nearly as much as the most able offspring of parents at the bottom of this range Then the conditional ability distribution of those earning incomes slightly greater than that earned by the most able children of the least well-off parents may be dominated by the conditional distribution of ability of those whose incomes are slightly less than the earnings of the least able children of the most well-off parents An example along these lines is discussed in Section 6

5 AN EXAMPLE

It is interesting to examine the explicit solution of this problem in the special case of Cobb-Douglas production and utility functions For the analysis of this section we assume that

h ( a e ) = (ae)

f ( a ) 3 1 a E [0 1 1

With these particular functions it is not hard to see that the consistent indirect utility function and optimal investment function have the following forms

(10) V ( y ) = k y s k gt 0 0 lt 6 lt 1 and

e (y ) = sy 0 lt s lt 1

The reader may verify by direct substitution that the functions V and e given above satisfy (2)when s = ( 1 + y)2 6 = 2 y ( l + y) and

By Theorem 1 we know these solutions are unique Thus in this simple instance we obtain a constant proportional investment

function This result continues to hold for arbitrary distributions of ability f and

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

848 GLENN C LOURY

(iii) There exists p gt 0 I gt 0 such that

ah e gt I max -(a e) 9 p lt 1 a ~ [ o ae11

ASSUMPTION3 Innate economic ability is distributed on the unit interval inde- pendently and identically for all agents The distribution has a continuous strictly positive density function f [0 11 +R+

3 CONSISTENT INDIRECT UTILITY AND THE DYNAMICS OF THE INCOME

DISTRIBUTION

Our assumptions (hereafter Al-A3 etc) enable us to demonstrate the exist- ence of a unique consistent indirect utility function V(y) and to make rather strong statements about the long run behavior of the distribution of earnings implied by equation (3) These essentially mathematical results are required before we can proceed to the economic analysis which is developed in the following section To see that the notion of consistent indirect utility is not vacuous we adopt to our context a dynamic programming argument originated by Bellman First however notice that A2(iii) implies there is a unique earnings level y such that h(1J) = y and y gty h(1 y ) lty Hence we may without loss of generality restrict our attention to incomes in the interval [0 j ] since no familys income could persist outside this range Let C [0 j ] denote the continu- ous real valued functions on [0 y] For 9 E C[O j ] define the function T+() as follows

(4) vy euro [0 y] T+(y) Eo7a E U(c 9(h(a Y - ~ 1 ) ) c y

An indirect utility function is consistent if and only if it is a fixed point of the mapping T Our assumptions permit us to show that T C [0 J] C [0 j ] is a contraction mapping and hence has a unique fixed point In Section 6 we prove the following

THEOREM1 Under Al A2 and A3 for the above defined positive number j there exists a unique consistent indirect utility function V on [0 y] V is strictly increasing strictly concave and differentiable on (0 y] The optimal consumption function c(y) is continuous and satisfies 0 lt c(y) lty Vy E (0 y] (Denote e(y) =y - c(y))

The property of uniqueness established here is very important For neither positive nor normative analysis would our story carry much force were we required to select arbitrarily among several consistent representations of family preferences On the other hand we have assumed a cardinal representation of preferences This is unavoidable if parents are to be taken as contemplating the welfare of their offspring While an axiomatic justification of our choice criterion might be sought along the lines of Koopmans [17] this task is complicated by the

849 INTERGENERATIONAL TRANSFERS

uncertainty inherent in our set-up and is not pursued here As one might expect V inherits the property of risk aversion from U This is important for much of the discussion of Section 4

We turn now to the study of the intergenerational motion of the distribution of income In any period (say nth) the distribution of family income is conceived as a probability measure on [0 j ] v The interpretation is that for A c [0 J] vn(A) is the fraction of all mature agents whose incomes lie in the set A In what follows we denote by 9 the set of probability measures on 9the Bore1 sets of [0 j ] p will denote the Lebesgue measure on 9

The system of family decision making outlined above enables us to treat income dynamics in this economy as a Markoff process To do so we require the following definition

DEFINITION1 A transition probability on [0 j ] is a function P [0 j ] x +[O 11 satisfying (i) Vy E [0 j ] P(y a ) E 9 (ii) VA E P( A) is a -measurable function on R+ To each transition probability on [0 j ] there corresponds a Markoff process That is once a transition probability has been specified one can generate for any fixed v E9a sequence of measures v) as follows

Thus the intergenerational motion of the distribution of income will be fully characterized once we have found the mechanism which relates the probability distribution of an offsprings income to the earnings of its parent Towards this end define the function h - x R++ as follows

Then it is clear that the probability an offspring has income in A when parents earnings is y is simply the probability that the offspring has an innate endow- ment in the set hK1(~e(y)) The relevant transition probability is given by

With P defined as in (6) it is obvious that P(y ) E9moreover it follows from a result of ~ u t i a that P ( A) is -measurable when e() is continuous The last fact was shown in Theorem 1

We now introduce the notion of an equilibrium distribution

DEFINITION2 An equilibrium income distribution is a measure v E 9 satisfy-ing VA E

See Futia [12 Theorem 52 p 741

850 GLENN C LOURY

Thus an equilibrium distribution is one which if it ever characterizes the distribution of earnings among a given generation continues to do so for each subsequent generation Assumptions A1-A3 already permit us to assert the existence of at least one equilibrium and convergence to some equilibrium However since our focus here is on dynamics it seems preferable to introduce the following simple assumption from which follows the global stability of the unique equilibrium point

That is education is a normal good We now have the following theorem

THEOREM2 Under Al-A4 there exists a unique equilibrium distribution v E 9If v) is a sequence of income distributions originating from the arbitrary initial v then limnvn = v pointwise on 9Moreover v has support [OA where y satisfies h (1 e($) =y

Thus by the proof given in Section 6 we establish existence uniqueness and stability for our notion of equilibrium Together with Theorem 1 this constitutes the framework within which we explore the issues raised in the introduction This examination is conducted in the next two sections

4 ECONOMIC ANALYSIS OF THE EQUILIBRIUM DISTRIBUTION

We have now constructed a relatively simple maximizing model of the distribu- tion of lifetime earnings among successive generations of workers The salient features of this model are (i) the recursive family preference structure reflecting intergenerational altruism (ii) the capital-theoretic characterization of produc- tion possibilities (iii) the stochastic nature of innate endowments assumed independent within each family over time and across families at any moment of time and (iv) the assumed absence of markets for human capital loans and income risk sharing between families These factors determine the average level of output in equilibrium and its dispersion as well as the criterion which families use to evaluate their wellbeing

Thus we may think of our model as defining a function M which associates with every triple of utility production and ability distribution functions satisfy- ing our assumptions a unique pair the equilibrium earnings distribution and consistent indirect utility

M (Uh f ) -+(v V)

A natural way to proceed in the analysis of this model is to ask comparative statics questions regarding how the endogenous (v V) vary with changes in the exogenous (U h f) This is a very difficult question in general though we are able to find some preliminary results along these lines presented below

Before proceeding to this however let us notice how the model distinguishes the phenomenon of social mobility across generations from that of inequality in

INTERGENERATIONAL TRANSFERS 851

earnings within a generation The relationship between these alternative notions of social stratification has been the focus of some attention in the literature [4 271 Here the two notions are inextricably intertwined intergenerational social mobil- ity is a property of the transition probability P while cross-sectional inequality is (asymptotically) a property of the equilibrium distribution to which P gives rise The basic problem for a normative analysis of economic inequality raised by this joint determination is that the concept of a more equal income distribution is not easily defined For instance how is one to compare a situation in which there is only a slight degree of inequality among families in any given generation but no mobility within families across generations with a circumstance in which there is substantial intragenerational income dispersion but also a large degree of intergenerational mobility Which situation evidences less inequality In the first case such inequality as exists between families lasts forever while in the latter case family members of a particular cohort experience significant earnings differences but no family is permanently assigned to the bottom or top of the earnings hierarchy

The formulation put forward here provides a way of thinking about this trade-off Our equilibrium notion depicts the intragenerational dispersion in earnings which would persist in the long run On the other hand the consistent indirect utility function incorporates into the valuation which individuals affix to any particular location in the income hierarchy the consequences of subsequent social mobility As noted earlier a great deal of intergenerational mobility implies a relatively flat indirect utility function making the cross-sectional distribution of welfare in equilibrium less unequal than would be the case with little or no mobility Thus by examining the distribution in equilibrium of consistent indirect utility (instead of income) one may simultaneously account for both kinds of inequality within a unified framework Indeed the statistic

is a natural extension of the standard utilitarian criterion and provides a complete ordering of social structures (U h f)satisfying A1-A4

Let us consider now the absence of loan and insurance markets in this model These market imperfections have important allocative consequences A number of writers have called attention to the fact that efficient resource allocation requires that children in low-income families should not be restricted by limited parental resources in their access to training For example Arthur Okun has remarked [22 p 80-811 concerning the modern US economy that The most important consequence (of an imperfect loan market) is the inadequate development of the human resources of the children of poor families-which would judge is one of the most serious inefficiencies of the American Economy today (Emphasis added) While it is certainly the case that in most societies a variety of devices exist for overcoming this problem they are far from perfect For example early childhood investments in nutrition or pre-school education are fundamentally income-constrained Nor should we expect a competitive loan

I

852 GLENN C LOURY

market to completely eliminate the dispersion in expected rates of return to training across families which arises in this model Legally poor parents will not be able to constrain their children to honor debts incurred on their behalf Nor will the newly matured children of wealthy families be able to attach the (human) assets of their less well-off counterparts should the latter decide for whatever reasons not to repay their loans (Default has been a pervasive problem with government guaranteed educational loan programs which would not exist absent public underwriting) Moreover the ability to make use of human capital is unknown even to the borrower Thus genuine bankruptcies can be expected as people mature to learn they are not as able as their parents gambled they would be The absence of inter-family loans in this model reflects an important feature of reality the allocative implications of which deserve study

Another critical feature of this story is that parents bear theoretically insurable risk when investing in their offspring Earnings of children are random here because the economic ability of the child becomes known only in maturity Yet a significant spreading of risk is technologically possible for a group of parents investing at the same level since their childrens earnings are independent identically distributed random variables Again we may expect that markets will fail to bring about complete risk spreading the moral hazard problems of income insurance contracts are o b v i ~ u s ~

Any attempt by a central authority to correct for these market imperfections must cope with the same incentive problems which limit the scope of the competitive mechanism here Efforts by a government to alter the allocation of training among young people will affect the consumption-investment decision of parents Moreover parents may be presumed to have better information than a central authority about the distribution of their childrens abilities (This is especially so if ability is positively correlated across generations of the same family) Yet as long as a group of parents with the same information have different incomes and there is no capital market their investment decisions will vary causing inefficiency This inefficiency could be reduced if the government were to redistribute incomes more equally but only at the cost of the excess burden accompanying such a redistributive tax

However because we have assumed altruism between generations a program of income redistribution which is imposed permanently has welfare effects here not encountered in the usual tax analysis The fact that income will be redistri- buted by taxing the next generation of workers causes each parent to regard his offsprings random income as less risky Since parents are risk averse egalitarian redistributive measures have certain welfare enhancing insurance characteris-tics It may be shown that as long as the redistributive activity is not too extensive this insurance effect dominates the excess burden effect Under such circumstances a permanent redistributive tax policy can be designed in such a way as to make all current mature agents better off

See Zeckhauser [30]for a discussion of these problems

853 INTERGENERATIONAL TRANSFERS

DEFINITION3 An education specific tax policy (ESTP) is a function r R + R+ with the interpretation that r(ye) is the after-tax income of someone whose income is y and training is e An ESTP is admissible if h(a e) = r(h(a e) e) satisfies Assumption 2 An admissible ESTP is purely redistributive (PRESTP) if j[r(h(a e) e) - h ( a e)]f(a)p (da) = 0 for every e gt 0 Denote by C the class of admissible PRESTP C is a non-empty convex set

DEFINITION4 The partial ordering is more egalitarian than denoted + is defined on C by the requirement 7 3 7 if and only if for each e the after tax distribution of income under 7 is more risky in the sense of second order stochastic dominance (see for example [24]) than the after tax distribution under

These two definitions allow us to formalize the discussion above By a result of Atkinson [I] 7 3 7 implies the Lorenze curve of the after tax income distribu- tion for persons of a given level of training under 7 lies on or above the corresponding curve under 7 However since the investment function e() will depend on 7 the same statement will not generally hold for the overall distribu- tions of income Thus we must use the term more egalitarian with some caution Let (y e) E jh(a e)f(a)p (da) and ~ ( y e) = y for all (y e) E

2+relative to Crepresenting perfect income insurance is maximal in Then on the other hand represents laisset faire For B E (0 I ) let r = B7 + (1 - )I Then r E C Finally for every r E C let V denote the unique solution for (2) when h is replaced by h

THEOREM3 Under A1-A3 7 $ r2 V(y) gt V(y) Vy gt 0 T1T2E C

Theorem 3 states that the use of a more egalitarian PRESTP among future generations of workers will lead to a higher level of welfare for each mature individual currently The careful reader will notice that this result depends upon our assumption that labor supply is exogenous In general we may expect that an income tax will distort both the educational investment decisions of parents and their choices concerning work effort By allowing the income tax function to vary with the level of educational investment (redistributing income within education classes only) we avoid the negative welfare consequences of the former distor- tion It is possible to preserve the result in Theorem 3 in the face of the latter kind of distortion as well if we restrict our attention to small redistributive efforts To see this imagine that h(ae) determines output per unit of time worked that each parent is endowed with one unit of time and that parents utility depends on work effort as well as consumption and offsprings welfare Under these slightly more general circumstances one can repeat the development of Theorems 1 and 2 with the natural modifications that indirect utility becomes a function of output per unit time (instead of total income) and income is the product of work effort and h(a e) Now consider introducing the tax scheme r (described above) for small B For fixed labor supply it may be shown that

854 GLENN C LOURY

welfare at any level of income is a concave function of B Moreover Theorem 3 implies that welfare is an increasing function of B However from standard tax theory we know that the excess burden of the tax T~ is zero when B = 0 and has derivative with respect to equal to zero at B = 0 (That is the excess burden of a small tax is a second order effect) Thus for sufficiently close to zero the insurance effect described in Theorem 3 will outweigh the excess burden effect of the tax scheme leading to an increase in wellbeing for all parents of the current generation

This result suggests that the standard tradeoff between equity and efficiency is somewhat less severe in the second-best world where opportunities to acquire training vary with parental resources It is sometimes possible to design policies which increase both equity and efficiency As a further indication of this last point let us examine the impact of public education in this model By public education we mean the centralized provision of training on an equal basis to every young person in the economy For the purposes of comparison we contrast the laissez faire outcome with that which occurs under public education with a per-capita budget equal to the expenditure of the average family in the no-intervention equilibrium We assume for this result that education is financed with non-distortionary tax

ASSUMPTION is convex in y5 h(ae(y)) is concave in y and ah(ae(y))aa

Assumption 5 posits that the marginal product of social background declines as social background (ie parents income) improves but declines less rapidly for the more able We have proven in Section 6 the following result

THEOREM4 Under A1-A5 universal public education with a per-capita budget equal to the investment of the average family in the laissez faire equilibrium will produce an earnings distribution with lower variance and higher mean than that which obtains without intervention

Finally let us turn to an examination of the relationship between earnings and ability in equilibrium It is widely held that differences in ability provide ethical grounds for differences in rewards An often cited justification for the market determined distribution of economic advantage is that it gives greater rewards to those of greater abilities In our model such is not literally the case because opportunities for training vary with social origin Thus rewards correspond to productivity but not ability Productivity depends both on ability and on social background Nonetheless some sort of systematic positive relationship between earnings and ability might still be sought Imagine that we observe in equilibrium an economy of the type studied here Suppose that we can observe an agents income but neither his ability nor his parents income Now the pair (iux) of individual ability and income is a jointly random vector in the population A very weak definition of a meritocratic distribution of income is that iu and x are positively correlated Under this definition the distribution is meritocratic if a regression of income on ability would yield a positive coefficient in a large

855 INTERGENERATIONAL TRANSFERS

sample A more substantive restriction would require that if x gt x then pr[G gt a 1 x = x ] gt pr[G gt a I x = x2]for all a E [0 11 Under this stronger defi- nition of a meritocratic distribution of rewards we require that greater earnings imply a greater conditional probability that innate aptitude exceeds any pre- specified level The following theorem asserts that the economy studied here is generally weakly meritocratic but not strongly so

THEOREM5 Under A1-A4 the economy is weakly meritocratic However it is not generally true that the economy is strongly meritocratic

The reason that a strong meritocracy may fail to obtain can be easily seen Suppose there exists some range of parental incomes over which the investment schedule e(y) increases rapidly such that the least able offspring of parents at the top earn nearly as much as the most able offspring of parents at the bottom of this range Then the conditional ability distribution of those earning incomes slightly greater than that earned by the most able children of the least well-off parents may be dominated by the conditional distribution of ability of those whose incomes are slightly less than the earnings of the least able children of the most well-off parents An example along these lines is discussed in Section 6

5 AN EXAMPLE

It is interesting to examine the explicit solution of this problem in the special case of Cobb-Douglas production and utility functions For the analysis of this section we assume that

h ( a e ) = (ae)

f ( a ) 3 1 a E [0 1 1

With these particular functions it is not hard to see that the consistent indirect utility function and optimal investment function have the following forms

(10) V ( y ) = k y s k gt 0 0 lt 6 lt 1 and

e (y ) = sy 0 lt s lt 1

The reader may verify by direct substitution that the functions V and e given above satisfy (2)when s = ( 1 + y)2 6 = 2 y ( l + y) and

By Theorem 1 we know these solutions are unique Thus in this simple instance we obtain a constant proportional investment

function This result continues to hold for arbitrary distributions of ability f and

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

849 INTERGENERATIONAL TRANSFERS

uncertainty inherent in our set-up and is not pursued here As one might expect V inherits the property of risk aversion from U This is important for much of the discussion of Section 4

We turn now to the study of the intergenerational motion of the distribution of income In any period (say nth) the distribution of family income is conceived as a probability measure on [0 j ] v The interpretation is that for A c [0 J] vn(A) is the fraction of all mature agents whose incomes lie in the set A In what follows we denote by 9 the set of probability measures on 9the Bore1 sets of [0 j ] p will denote the Lebesgue measure on 9

The system of family decision making outlined above enables us to treat income dynamics in this economy as a Markoff process To do so we require the following definition

DEFINITION1 A transition probability on [0 j ] is a function P [0 j ] x +[O 11 satisfying (i) Vy E [0 j ] P(y a ) E 9 (ii) VA E P( A) is a -measurable function on R+ To each transition probability on [0 j ] there corresponds a Markoff process That is once a transition probability has been specified one can generate for any fixed v E9a sequence of measures v) as follows

Thus the intergenerational motion of the distribution of income will be fully characterized once we have found the mechanism which relates the probability distribution of an offsprings income to the earnings of its parent Towards this end define the function h - x R++ as follows

Then it is clear that the probability an offspring has income in A when parents earnings is y is simply the probability that the offspring has an innate endow- ment in the set hK1(~e(y)) The relevant transition probability is given by

With P defined as in (6) it is obvious that P(y ) E9moreover it follows from a result of ~ u t i a that P ( A) is -measurable when e() is continuous The last fact was shown in Theorem 1

We now introduce the notion of an equilibrium distribution

DEFINITION2 An equilibrium income distribution is a measure v E 9 satisfy-ing VA E

See Futia [12 Theorem 52 p 741

850 GLENN C LOURY

Thus an equilibrium distribution is one which if it ever characterizes the distribution of earnings among a given generation continues to do so for each subsequent generation Assumptions A1-A3 already permit us to assert the existence of at least one equilibrium and convergence to some equilibrium However since our focus here is on dynamics it seems preferable to introduce the following simple assumption from which follows the global stability of the unique equilibrium point

That is education is a normal good We now have the following theorem

THEOREM2 Under Al-A4 there exists a unique equilibrium distribution v E 9If v) is a sequence of income distributions originating from the arbitrary initial v then limnvn = v pointwise on 9Moreover v has support [OA where y satisfies h (1 e($) =y

Thus by the proof given in Section 6 we establish existence uniqueness and stability for our notion of equilibrium Together with Theorem 1 this constitutes the framework within which we explore the issues raised in the introduction This examination is conducted in the next two sections

4 ECONOMIC ANALYSIS OF THE EQUILIBRIUM DISTRIBUTION

We have now constructed a relatively simple maximizing model of the distribu- tion of lifetime earnings among successive generations of workers The salient features of this model are (i) the recursive family preference structure reflecting intergenerational altruism (ii) the capital-theoretic characterization of produc- tion possibilities (iii) the stochastic nature of innate endowments assumed independent within each family over time and across families at any moment of time and (iv) the assumed absence of markets for human capital loans and income risk sharing between families These factors determine the average level of output in equilibrium and its dispersion as well as the criterion which families use to evaluate their wellbeing

Thus we may think of our model as defining a function M which associates with every triple of utility production and ability distribution functions satisfy- ing our assumptions a unique pair the equilibrium earnings distribution and consistent indirect utility

M (Uh f ) -+(v V)

A natural way to proceed in the analysis of this model is to ask comparative statics questions regarding how the endogenous (v V) vary with changes in the exogenous (U h f) This is a very difficult question in general though we are able to find some preliminary results along these lines presented below

Before proceeding to this however let us notice how the model distinguishes the phenomenon of social mobility across generations from that of inequality in

INTERGENERATIONAL TRANSFERS 851

earnings within a generation The relationship between these alternative notions of social stratification has been the focus of some attention in the literature [4 271 Here the two notions are inextricably intertwined intergenerational social mobil- ity is a property of the transition probability P while cross-sectional inequality is (asymptotically) a property of the equilibrium distribution to which P gives rise The basic problem for a normative analysis of economic inequality raised by this joint determination is that the concept of a more equal income distribution is not easily defined For instance how is one to compare a situation in which there is only a slight degree of inequality among families in any given generation but no mobility within families across generations with a circumstance in which there is substantial intragenerational income dispersion but also a large degree of intergenerational mobility Which situation evidences less inequality In the first case such inequality as exists between families lasts forever while in the latter case family members of a particular cohort experience significant earnings differences but no family is permanently assigned to the bottom or top of the earnings hierarchy

The formulation put forward here provides a way of thinking about this trade-off Our equilibrium notion depicts the intragenerational dispersion in earnings which would persist in the long run On the other hand the consistent indirect utility function incorporates into the valuation which individuals affix to any particular location in the income hierarchy the consequences of subsequent social mobility As noted earlier a great deal of intergenerational mobility implies a relatively flat indirect utility function making the cross-sectional distribution of welfare in equilibrium less unequal than would be the case with little or no mobility Thus by examining the distribution in equilibrium of consistent indirect utility (instead of income) one may simultaneously account for both kinds of inequality within a unified framework Indeed the statistic

is a natural extension of the standard utilitarian criterion and provides a complete ordering of social structures (U h f)satisfying A1-A4

Let us consider now the absence of loan and insurance markets in this model These market imperfections have important allocative consequences A number of writers have called attention to the fact that efficient resource allocation requires that children in low-income families should not be restricted by limited parental resources in their access to training For example Arthur Okun has remarked [22 p 80-811 concerning the modern US economy that The most important consequence (of an imperfect loan market) is the inadequate development of the human resources of the children of poor families-which would judge is one of the most serious inefficiencies of the American Economy today (Emphasis added) While it is certainly the case that in most societies a variety of devices exist for overcoming this problem they are far from perfect For example early childhood investments in nutrition or pre-school education are fundamentally income-constrained Nor should we expect a competitive loan

I

852 GLENN C LOURY

market to completely eliminate the dispersion in expected rates of return to training across families which arises in this model Legally poor parents will not be able to constrain their children to honor debts incurred on their behalf Nor will the newly matured children of wealthy families be able to attach the (human) assets of their less well-off counterparts should the latter decide for whatever reasons not to repay their loans (Default has been a pervasive problem with government guaranteed educational loan programs which would not exist absent public underwriting) Moreover the ability to make use of human capital is unknown even to the borrower Thus genuine bankruptcies can be expected as people mature to learn they are not as able as their parents gambled they would be The absence of inter-family loans in this model reflects an important feature of reality the allocative implications of which deserve study

Another critical feature of this story is that parents bear theoretically insurable risk when investing in their offspring Earnings of children are random here because the economic ability of the child becomes known only in maturity Yet a significant spreading of risk is technologically possible for a group of parents investing at the same level since their childrens earnings are independent identically distributed random variables Again we may expect that markets will fail to bring about complete risk spreading the moral hazard problems of income insurance contracts are o b v i ~ u s ~

Any attempt by a central authority to correct for these market imperfections must cope with the same incentive problems which limit the scope of the competitive mechanism here Efforts by a government to alter the allocation of training among young people will affect the consumption-investment decision of parents Moreover parents may be presumed to have better information than a central authority about the distribution of their childrens abilities (This is especially so if ability is positively correlated across generations of the same family) Yet as long as a group of parents with the same information have different incomes and there is no capital market their investment decisions will vary causing inefficiency This inefficiency could be reduced if the government were to redistribute incomes more equally but only at the cost of the excess burden accompanying such a redistributive tax

However because we have assumed altruism between generations a program of income redistribution which is imposed permanently has welfare effects here not encountered in the usual tax analysis The fact that income will be redistri- buted by taxing the next generation of workers causes each parent to regard his offsprings random income as less risky Since parents are risk averse egalitarian redistributive measures have certain welfare enhancing insurance characteris-tics It may be shown that as long as the redistributive activity is not too extensive this insurance effect dominates the excess burden effect Under such circumstances a permanent redistributive tax policy can be designed in such a way as to make all current mature agents better off

See Zeckhauser [30]for a discussion of these problems

853 INTERGENERATIONAL TRANSFERS

DEFINITION3 An education specific tax policy (ESTP) is a function r R + R+ with the interpretation that r(ye) is the after-tax income of someone whose income is y and training is e An ESTP is admissible if h(a e) = r(h(a e) e) satisfies Assumption 2 An admissible ESTP is purely redistributive (PRESTP) if j[r(h(a e) e) - h ( a e)]f(a)p (da) = 0 for every e gt 0 Denote by C the class of admissible PRESTP C is a non-empty convex set

DEFINITION4 The partial ordering is more egalitarian than denoted + is defined on C by the requirement 7 3 7 if and only if for each e the after tax distribution of income under 7 is more risky in the sense of second order stochastic dominance (see for example [24]) than the after tax distribution under

These two definitions allow us to formalize the discussion above By a result of Atkinson [I] 7 3 7 implies the Lorenze curve of the after tax income distribu- tion for persons of a given level of training under 7 lies on or above the corresponding curve under 7 However since the investment function e() will depend on 7 the same statement will not generally hold for the overall distribu- tions of income Thus we must use the term more egalitarian with some caution Let (y e) E jh(a e)f(a)p (da) and ~ ( y e) = y for all (y e) E

2+relative to Crepresenting perfect income insurance is maximal in Then on the other hand represents laisset faire For B E (0 I ) let r = B7 + (1 - )I Then r E C Finally for every r E C let V denote the unique solution for (2) when h is replaced by h

THEOREM3 Under A1-A3 7 $ r2 V(y) gt V(y) Vy gt 0 T1T2E C

Theorem 3 states that the use of a more egalitarian PRESTP among future generations of workers will lead to a higher level of welfare for each mature individual currently The careful reader will notice that this result depends upon our assumption that labor supply is exogenous In general we may expect that an income tax will distort both the educational investment decisions of parents and their choices concerning work effort By allowing the income tax function to vary with the level of educational investment (redistributing income within education classes only) we avoid the negative welfare consequences of the former distor- tion It is possible to preserve the result in Theorem 3 in the face of the latter kind of distortion as well if we restrict our attention to small redistributive efforts To see this imagine that h(ae) determines output per unit of time worked that each parent is endowed with one unit of time and that parents utility depends on work effort as well as consumption and offsprings welfare Under these slightly more general circumstances one can repeat the development of Theorems 1 and 2 with the natural modifications that indirect utility becomes a function of output per unit time (instead of total income) and income is the product of work effort and h(a e) Now consider introducing the tax scheme r (described above) for small B For fixed labor supply it may be shown that

854 GLENN C LOURY

welfare at any level of income is a concave function of B Moreover Theorem 3 implies that welfare is an increasing function of B However from standard tax theory we know that the excess burden of the tax T~ is zero when B = 0 and has derivative with respect to equal to zero at B = 0 (That is the excess burden of a small tax is a second order effect) Thus for sufficiently close to zero the insurance effect described in Theorem 3 will outweigh the excess burden effect of the tax scheme leading to an increase in wellbeing for all parents of the current generation

This result suggests that the standard tradeoff between equity and efficiency is somewhat less severe in the second-best world where opportunities to acquire training vary with parental resources It is sometimes possible to design policies which increase both equity and efficiency As a further indication of this last point let us examine the impact of public education in this model By public education we mean the centralized provision of training on an equal basis to every young person in the economy For the purposes of comparison we contrast the laissez faire outcome with that which occurs under public education with a per-capita budget equal to the expenditure of the average family in the no-intervention equilibrium We assume for this result that education is financed with non-distortionary tax

ASSUMPTION is convex in y5 h(ae(y)) is concave in y and ah(ae(y))aa

Assumption 5 posits that the marginal product of social background declines as social background (ie parents income) improves but declines less rapidly for the more able We have proven in Section 6 the following result

THEOREM4 Under A1-A5 universal public education with a per-capita budget equal to the investment of the average family in the laissez faire equilibrium will produce an earnings distribution with lower variance and higher mean than that which obtains without intervention

Finally let us turn to an examination of the relationship between earnings and ability in equilibrium It is widely held that differences in ability provide ethical grounds for differences in rewards An often cited justification for the market determined distribution of economic advantage is that it gives greater rewards to those of greater abilities In our model such is not literally the case because opportunities for training vary with social origin Thus rewards correspond to productivity but not ability Productivity depends both on ability and on social background Nonetheless some sort of systematic positive relationship between earnings and ability might still be sought Imagine that we observe in equilibrium an economy of the type studied here Suppose that we can observe an agents income but neither his ability nor his parents income Now the pair (iux) of individual ability and income is a jointly random vector in the population A very weak definition of a meritocratic distribution of income is that iu and x are positively correlated Under this definition the distribution is meritocratic if a regression of income on ability would yield a positive coefficient in a large

855 INTERGENERATIONAL TRANSFERS

sample A more substantive restriction would require that if x gt x then pr[G gt a 1 x = x ] gt pr[G gt a I x = x2]for all a E [0 11 Under this stronger defi- nition of a meritocratic distribution of rewards we require that greater earnings imply a greater conditional probability that innate aptitude exceeds any pre- specified level The following theorem asserts that the economy studied here is generally weakly meritocratic but not strongly so

THEOREM5 Under A1-A4 the economy is weakly meritocratic However it is not generally true that the economy is strongly meritocratic

The reason that a strong meritocracy may fail to obtain can be easily seen Suppose there exists some range of parental incomes over which the investment schedule e(y) increases rapidly such that the least able offspring of parents at the top earn nearly as much as the most able offspring of parents at the bottom of this range Then the conditional ability distribution of those earning incomes slightly greater than that earned by the most able children of the least well-off parents may be dominated by the conditional distribution of ability of those whose incomes are slightly less than the earnings of the least able children of the most well-off parents An example along these lines is discussed in Section 6

5 AN EXAMPLE

It is interesting to examine the explicit solution of this problem in the special case of Cobb-Douglas production and utility functions For the analysis of this section we assume that

h ( a e ) = (ae)

f ( a ) 3 1 a E [0 1 1

With these particular functions it is not hard to see that the consistent indirect utility function and optimal investment function have the following forms

(10) V ( y ) = k y s k gt 0 0 lt 6 lt 1 and

e (y ) = sy 0 lt s lt 1

The reader may verify by direct substitution that the functions V and e given above satisfy (2)when s = ( 1 + y)2 6 = 2 y ( l + y) and

By Theorem 1 we know these solutions are unique Thus in this simple instance we obtain a constant proportional investment

function This result continues to hold for arbitrary distributions of ability f and

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

850 GLENN C LOURY

Thus an equilibrium distribution is one which if it ever characterizes the distribution of earnings among a given generation continues to do so for each subsequent generation Assumptions A1-A3 already permit us to assert the existence of at least one equilibrium and convergence to some equilibrium However since our focus here is on dynamics it seems preferable to introduce the following simple assumption from which follows the global stability of the unique equilibrium point

That is education is a normal good We now have the following theorem

THEOREM2 Under Al-A4 there exists a unique equilibrium distribution v E 9If v) is a sequence of income distributions originating from the arbitrary initial v then limnvn = v pointwise on 9Moreover v has support [OA where y satisfies h (1 e($) =y

Thus by the proof given in Section 6 we establish existence uniqueness and stability for our notion of equilibrium Together with Theorem 1 this constitutes the framework within which we explore the issues raised in the introduction This examination is conducted in the next two sections

4 ECONOMIC ANALYSIS OF THE EQUILIBRIUM DISTRIBUTION

We have now constructed a relatively simple maximizing model of the distribu- tion of lifetime earnings among successive generations of workers The salient features of this model are (i) the recursive family preference structure reflecting intergenerational altruism (ii) the capital-theoretic characterization of produc- tion possibilities (iii) the stochastic nature of innate endowments assumed independent within each family over time and across families at any moment of time and (iv) the assumed absence of markets for human capital loans and income risk sharing between families These factors determine the average level of output in equilibrium and its dispersion as well as the criterion which families use to evaluate their wellbeing

Thus we may think of our model as defining a function M which associates with every triple of utility production and ability distribution functions satisfy- ing our assumptions a unique pair the equilibrium earnings distribution and consistent indirect utility

M (Uh f ) -+(v V)

A natural way to proceed in the analysis of this model is to ask comparative statics questions regarding how the endogenous (v V) vary with changes in the exogenous (U h f) This is a very difficult question in general though we are able to find some preliminary results along these lines presented below

Before proceeding to this however let us notice how the model distinguishes the phenomenon of social mobility across generations from that of inequality in

INTERGENERATIONAL TRANSFERS 851

earnings within a generation The relationship between these alternative notions of social stratification has been the focus of some attention in the literature [4 271 Here the two notions are inextricably intertwined intergenerational social mobil- ity is a property of the transition probability P while cross-sectional inequality is (asymptotically) a property of the equilibrium distribution to which P gives rise The basic problem for a normative analysis of economic inequality raised by this joint determination is that the concept of a more equal income distribution is not easily defined For instance how is one to compare a situation in which there is only a slight degree of inequality among families in any given generation but no mobility within families across generations with a circumstance in which there is substantial intragenerational income dispersion but also a large degree of intergenerational mobility Which situation evidences less inequality In the first case such inequality as exists between families lasts forever while in the latter case family members of a particular cohort experience significant earnings differences but no family is permanently assigned to the bottom or top of the earnings hierarchy

The formulation put forward here provides a way of thinking about this trade-off Our equilibrium notion depicts the intragenerational dispersion in earnings which would persist in the long run On the other hand the consistent indirect utility function incorporates into the valuation which individuals affix to any particular location in the income hierarchy the consequences of subsequent social mobility As noted earlier a great deal of intergenerational mobility implies a relatively flat indirect utility function making the cross-sectional distribution of welfare in equilibrium less unequal than would be the case with little or no mobility Thus by examining the distribution in equilibrium of consistent indirect utility (instead of income) one may simultaneously account for both kinds of inequality within a unified framework Indeed the statistic

is a natural extension of the standard utilitarian criterion and provides a complete ordering of social structures (U h f)satisfying A1-A4

Let us consider now the absence of loan and insurance markets in this model These market imperfections have important allocative consequences A number of writers have called attention to the fact that efficient resource allocation requires that children in low-income families should not be restricted by limited parental resources in their access to training For example Arthur Okun has remarked [22 p 80-811 concerning the modern US economy that The most important consequence (of an imperfect loan market) is the inadequate development of the human resources of the children of poor families-which would judge is one of the most serious inefficiencies of the American Economy today (Emphasis added) While it is certainly the case that in most societies a variety of devices exist for overcoming this problem they are far from perfect For example early childhood investments in nutrition or pre-school education are fundamentally income-constrained Nor should we expect a competitive loan

I

852 GLENN C LOURY

market to completely eliminate the dispersion in expected rates of return to training across families which arises in this model Legally poor parents will not be able to constrain their children to honor debts incurred on their behalf Nor will the newly matured children of wealthy families be able to attach the (human) assets of their less well-off counterparts should the latter decide for whatever reasons not to repay their loans (Default has been a pervasive problem with government guaranteed educational loan programs which would not exist absent public underwriting) Moreover the ability to make use of human capital is unknown even to the borrower Thus genuine bankruptcies can be expected as people mature to learn they are not as able as their parents gambled they would be The absence of inter-family loans in this model reflects an important feature of reality the allocative implications of which deserve study

Another critical feature of this story is that parents bear theoretically insurable risk when investing in their offspring Earnings of children are random here because the economic ability of the child becomes known only in maturity Yet a significant spreading of risk is technologically possible for a group of parents investing at the same level since their childrens earnings are independent identically distributed random variables Again we may expect that markets will fail to bring about complete risk spreading the moral hazard problems of income insurance contracts are o b v i ~ u s ~

Any attempt by a central authority to correct for these market imperfections must cope with the same incentive problems which limit the scope of the competitive mechanism here Efforts by a government to alter the allocation of training among young people will affect the consumption-investment decision of parents Moreover parents may be presumed to have better information than a central authority about the distribution of their childrens abilities (This is especially so if ability is positively correlated across generations of the same family) Yet as long as a group of parents with the same information have different incomes and there is no capital market their investment decisions will vary causing inefficiency This inefficiency could be reduced if the government were to redistribute incomes more equally but only at the cost of the excess burden accompanying such a redistributive tax

However because we have assumed altruism between generations a program of income redistribution which is imposed permanently has welfare effects here not encountered in the usual tax analysis The fact that income will be redistri- buted by taxing the next generation of workers causes each parent to regard his offsprings random income as less risky Since parents are risk averse egalitarian redistributive measures have certain welfare enhancing insurance characteris-tics It may be shown that as long as the redistributive activity is not too extensive this insurance effect dominates the excess burden effect Under such circumstances a permanent redistributive tax policy can be designed in such a way as to make all current mature agents better off

See Zeckhauser [30]for a discussion of these problems

853 INTERGENERATIONAL TRANSFERS

DEFINITION3 An education specific tax policy (ESTP) is a function r R + R+ with the interpretation that r(ye) is the after-tax income of someone whose income is y and training is e An ESTP is admissible if h(a e) = r(h(a e) e) satisfies Assumption 2 An admissible ESTP is purely redistributive (PRESTP) if j[r(h(a e) e) - h ( a e)]f(a)p (da) = 0 for every e gt 0 Denote by C the class of admissible PRESTP C is a non-empty convex set

DEFINITION4 The partial ordering is more egalitarian than denoted + is defined on C by the requirement 7 3 7 if and only if for each e the after tax distribution of income under 7 is more risky in the sense of second order stochastic dominance (see for example [24]) than the after tax distribution under

These two definitions allow us to formalize the discussion above By a result of Atkinson [I] 7 3 7 implies the Lorenze curve of the after tax income distribu- tion for persons of a given level of training under 7 lies on or above the corresponding curve under 7 However since the investment function e() will depend on 7 the same statement will not generally hold for the overall distribu- tions of income Thus we must use the term more egalitarian with some caution Let (y e) E jh(a e)f(a)p (da) and ~ ( y e) = y for all (y e) E

2+relative to Crepresenting perfect income insurance is maximal in Then on the other hand represents laisset faire For B E (0 I ) let r = B7 + (1 - )I Then r E C Finally for every r E C let V denote the unique solution for (2) when h is replaced by h

THEOREM3 Under A1-A3 7 $ r2 V(y) gt V(y) Vy gt 0 T1T2E C

Theorem 3 states that the use of a more egalitarian PRESTP among future generations of workers will lead to a higher level of welfare for each mature individual currently The careful reader will notice that this result depends upon our assumption that labor supply is exogenous In general we may expect that an income tax will distort both the educational investment decisions of parents and their choices concerning work effort By allowing the income tax function to vary with the level of educational investment (redistributing income within education classes only) we avoid the negative welfare consequences of the former distor- tion It is possible to preserve the result in Theorem 3 in the face of the latter kind of distortion as well if we restrict our attention to small redistributive efforts To see this imagine that h(ae) determines output per unit of time worked that each parent is endowed with one unit of time and that parents utility depends on work effort as well as consumption and offsprings welfare Under these slightly more general circumstances one can repeat the development of Theorems 1 and 2 with the natural modifications that indirect utility becomes a function of output per unit time (instead of total income) and income is the product of work effort and h(a e) Now consider introducing the tax scheme r (described above) for small B For fixed labor supply it may be shown that

854 GLENN C LOURY

welfare at any level of income is a concave function of B Moreover Theorem 3 implies that welfare is an increasing function of B However from standard tax theory we know that the excess burden of the tax T~ is zero when B = 0 and has derivative with respect to equal to zero at B = 0 (That is the excess burden of a small tax is a second order effect) Thus for sufficiently close to zero the insurance effect described in Theorem 3 will outweigh the excess burden effect of the tax scheme leading to an increase in wellbeing for all parents of the current generation

This result suggests that the standard tradeoff between equity and efficiency is somewhat less severe in the second-best world where opportunities to acquire training vary with parental resources It is sometimes possible to design policies which increase both equity and efficiency As a further indication of this last point let us examine the impact of public education in this model By public education we mean the centralized provision of training on an equal basis to every young person in the economy For the purposes of comparison we contrast the laissez faire outcome with that which occurs under public education with a per-capita budget equal to the expenditure of the average family in the no-intervention equilibrium We assume for this result that education is financed with non-distortionary tax

ASSUMPTION is convex in y5 h(ae(y)) is concave in y and ah(ae(y))aa

Assumption 5 posits that the marginal product of social background declines as social background (ie parents income) improves but declines less rapidly for the more able We have proven in Section 6 the following result

THEOREM4 Under A1-A5 universal public education with a per-capita budget equal to the investment of the average family in the laissez faire equilibrium will produce an earnings distribution with lower variance and higher mean than that which obtains without intervention

Finally let us turn to an examination of the relationship between earnings and ability in equilibrium It is widely held that differences in ability provide ethical grounds for differences in rewards An often cited justification for the market determined distribution of economic advantage is that it gives greater rewards to those of greater abilities In our model such is not literally the case because opportunities for training vary with social origin Thus rewards correspond to productivity but not ability Productivity depends both on ability and on social background Nonetheless some sort of systematic positive relationship between earnings and ability might still be sought Imagine that we observe in equilibrium an economy of the type studied here Suppose that we can observe an agents income but neither his ability nor his parents income Now the pair (iux) of individual ability and income is a jointly random vector in the population A very weak definition of a meritocratic distribution of income is that iu and x are positively correlated Under this definition the distribution is meritocratic if a regression of income on ability would yield a positive coefficient in a large

855 INTERGENERATIONAL TRANSFERS

sample A more substantive restriction would require that if x gt x then pr[G gt a 1 x = x ] gt pr[G gt a I x = x2]for all a E [0 11 Under this stronger defi- nition of a meritocratic distribution of rewards we require that greater earnings imply a greater conditional probability that innate aptitude exceeds any pre- specified level The following theorem asserts that the economy studied here is generally weakly meritocratic but not strongly so

THEOREM5 Under A1-A4 the economy is weakly meritocratic However it is not generally true that the economy is strongly meritocratic

The reason that a strong meritocracy may fail to obtain can be easily seen Suppose there exists some range of parental incomes over which the investment schedule e(y) increases rapidly such that the least able offspring of parents at the top earn nearly as much as the most able offspring of parents at the bottom of this range Then the conditional ability distribution of those earning incomes slightly greater than that earned by the most able children of the least well-off parents may be dominated by the conditional distribution of ability of those whose incomes are slightly less than the earnings of the least able children of the most well-off parents An example along these lines is discussed in Section 6

5 AN EXAMPLE

It is interesting to examine the explicit solution of this problem in the special case of Cobb-Douglas production and utility functions For the analysis of this section we assume that

h ( a e ) = (ae)

f ( a ) 3 1 a E [0 1 1

With these particular functions it is not hard to see that the consistent indirect utility function and optimal investment function have the following forms

(10) V ( y ) = k y s k gt 0 0 lt 6 lt 1 and

e (y ) = sy 0 lt s lt 1

The reader may verify by direct substitution that the functions V and e given above satisfy (2)when s = ( 1 + y)2 6 = 2 y ( l + y) and

By Theorem 1 we know these solutions are unique Thus in this simple instance we obtain a constant proportional investment

function This result continues to hold for arbitrary distributions of ability f and

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

INTERGENERATIONAL TRANSFERS 851

earnings within a generation The relationship between these alternative notions of social stratification has been the focus of some attention in the literature [4 271 Here the two notions are inextricably intertwined intergenerational social mobil- ity is a property of the transition probability P while cross-sectional inequality is (asymptotically) a property of the equilibrium distribution to which P gives rise The basic problem for a normative analysis of economic inequality raised by this joint determination is that the concept of a more equal income distribution is not easily defined For instance how is one to compare a situation in which there is only a slight degree of inequality among families in any given generation but no mobility within families across generations with a circumstance in which there is substantial intragenerational income dispersion but also a large degree of intergenerational mobility Which situation evidences less inequality In the first case such inequality as exists between families lasts forever while in the latter case family members of a particular cohort experience significant earnings differences but no family is permanently assigned to the bottom or top of the earnings hierarchy

The formulation put forward here provides a way of thinking about this trade-off Our equilibrium notion depicts the intragenerational dispersion in earnings which would persist in the long run On the other hand the consistent indirect utility function incorporates into the valuation which individuals affix to any particular location in the income hierarchy the consequences of subsequent social mobility As noted earlier a great deal of intergenerational mobility implies a relatively flat indirect utility function making the cross-sectional distribution of welfare in equilibrium less unequal than would be the case with little or no mobility Thus by examining the distribution in equilibrium of consistent indirect utility (instead of income) one may simultaneously account for both kinds of inequality within a unified framework Indeed the statistic

is a natural extension of the standard utilitarian criterion and provides a complete ordering of social structures (U h f)satisfying A1-A4

Let us consider now the absence of loan and insurance markets in this model These market imperfections have important allocative consequences A number of writers have called attention to the fact that efficient resource allocation requires that children in low-income families should not be restricted by limited parental resources in their access to training For example Arthur Okun has remarked [22 p 80-811 concerning the modern US economy that The most important consequence (of an imperfect loan market) is the inadequate development of the human resources of the children of poor families-which would judge is one of the most serious inefficiencies of the American Economy today (Emphasis added) While it is certainly the case that in most societies a variety of devices exist for overcoming this problem they are far from perfect For example early childhood investments in nutrition or pre-school education are fundamentally income-constrained Nor should we expect a competitive loan

I

852 GLENN C LOURY

market to completely eliminate the dispersion in expected rates of return to training across families which arises in this model Legally poor parents will not be able to constrain their children to honor debts incurred on their behalf Nor will the newly matured children of wealthy families be able to attach the (human) assets of their less well-off counterparts should the latter decide for whatever reasons not to repay their loans (Default has been a pervasive problem with government guaranteed educational loan programs which would not exist absent public underwriting) Moreover the ability to make use of human capital is unknown even to the borrower Thus genuine bankruptcies can be expected as people mature to learn they are not as able as their parents gambled they would be The absence of inter-family loans in this model reflects an important feature of reality the allocative implications of which deserve study

Another critical feature of this story is that parents bear theoretically insurable risk when investing in their offspring Earnings of children are random here because the economic ability of the child becomes known only in maturity Yet a significant spreading of risk is technologically possible for a group of parents investing at the same level since their childrens earnings are independent identically distributed random variables Again we may expect that markets will fail to bring about complete risk spreading the moral hazard problems of income insurance contracts are o b v i ~ u s ~

Any attempt by a central authority to correct for these market imperfections must cope with the same incentive problems which limit the scope of the competitive mechanism here Efforts by a government to alter the allocation of training among young people will affect the consumption-investment decision of parents Moreover parents may be presumed to have better information than a central authority about the distribution of their childrens abilities (This is especially so if ability is positively correlated across generations of the same family) Yet as long as a group of parents with the same information have different incomes and there is no capital market their investment decisions will vary causing inefficiency This inefficiency could be reduced if the government were to redistribute incomes more equally but only at the cost of the excess burden accompanying such a redistributive tax

However because we have assumed altruism between generations a program of income redistribution which is imposed permanently has welfare effects here not encountered in the usual tax analysis The fact that income will be redistri- buted by taxing the next generation of workers causes each parent to regard his offsprings random income as less risky Since parents are risk averse egalitarian redistributive measures have certain welfare enhancing insurance characteris-tics It may be shown that as long as the redistributive activity is not too extensive this insurance effect dominates the excess burden effect Under such circumstances a permanent redistributive tax policy can be designed in such a way as to make all current mature agents better off

See Zeckhauser [30]for a discussion of these problems

853 INTERGENERATIONAL TRANSFERS

DEFINITION3 An education specific tax policy (ESTP) is a function r R + R+ with the interpretation that r(ye) is the after-tax income of someone whose income is y and training is e An ESTP is admissible if h(a e) = r(h(a e) e) satisfies Assumption 2 An admissible ESTP is purely redistributive (PRESTP) if j[r(h(a e) e) - h ( a e)]f(a)p (da) = 0 for every e gt 0 Denote by C the class of admissible PRESTP C is a non-empty convex set

DEFINITION4 The partial ordering is more egalitarian than denoted + is defined on C by the requirement 7 3 7 if and only if for each e the after tax distribution of income under 7 is more risky in the sense of second order stochastic dominance (see for example [24]) than the after tax distribution under

These two definitions allow us to formalize the discussion above By a result of Atkinson [I] 7 3 7 implies the Lorenze curve of the after tax income distribu- tion for persons of a given level of training under 7 lies on or above the corresponding curve under 7 However since the investment function e() will depend on 7 the same statement will not generally hold for the overall distribu- tions of income Thus we must use the term more egalitarian with some caution Let (y e) E jh(a e)f(a)p (da) and ~ ( y e) = y for all (y e) E

2+relative to Crepresenting perfect income insurance is maximal in Then on the other hand represents laisset faire For B E (0 I ) let r = B7 + (1 - )I Then r E C Finally for every r E C let V denote the unique solution for (2) when h is replaced by h

THEOREM3 Under A1-A3 7 $ r2 V(y) gt V(y) Vy gt 0 T1T2E C

Theorem 3 states that the use of a more egalitarian PRESTP among future generations of workers will lead to a higher level of welfare for each mature individual currently The careful reader will notice that this result depends upon our assumption that labor supply is exogenous In general we may expect that an income tax will distort both the educational investment decisions of parents and their choices concerning work effort By allowing the income tax function to vary with the level of educational investment (redistributing income within education classes only) we avoid the negative welfare consequences of the former distor- tion It is possible to preserve the result in Theorem 3 in the face of the latter kind of distortion as well if we restrict our attention to small redistributive efforts To see this imagine that h(ae) determines output per unit of time worked that each parent is endowed with one unit of time and that parents utility depends on work effort as well as consumption and offsprings welfare Under these slightly more general circumstances one can repeat the development of Theorems 1 and 2 with the natural modifications that indirect utility becomes a function of output per unit time (instead of total income) and income is the product of work effort and h(a e) Now consider introducing the tax scheme r (described above) for small B For fixed labor supply it may be shown that

854 GLENN C LOURY

welfare at any level of income is a concave function of B Moreover Theorem 3 implies that welfare is an increasing function of B However from standard tax theory we know that the excess burden of the tax T~ is zero when B = 0 and has derivative with respect to equal to zero at B = 0 (That is the excess burden of a small tax is a second order effect) Thus for sufficiently close to zero the insurance effect described in Theorem 3 will outweigh the excess burden effect of the tax scheme leading to an increase in wellbeing for all parents of the current generation

This result suggests that the standard tradeoff between equity and efficiency is somewhat less severe in the second-best world where opportunities to acquire training vary with parental resources It is sometimes possible to design policies which increase both equity and efficiency As a further indication of this last point let us examine the impact of public education in this model By public education we mean the centralized provision of training on an equal basis to every young person in the economy For the purposes of comparison we contrast the laissez faire outcome with that which occurs under public education with a per-capita budget equal to the expenditure of the average family in the no-intervention equilibrium We assume for this result that education is financed with non-distortionary tax

ASSUMPTION is convex in y5 h(ae(y)) is concave in y and ah(ae(y))aa

Assumption 5 posits that the marginal product of social background declines as social background (ie parents income) improves but declines less rapidly for the more able We have proven in Section 6 the following result

THEOREM4 Under A1-A5 universal public education with a per-capita budget equal to the investment of the average family in the laissez faire equilibrium will produce an earnings distribution with lower variance and higher mean than that which obtains without intervention

Finally let us turn to an examination of the relationship between earnings and ability in equilibrium It is widely held that differences in ability provide ethical grounds for differences in rewards An often cited justification for the market determined distribution of economic advantage is that it gives greater rewards to those of greater abilities In our model such is not literally the case because opportunities for training vary with social origin Thus rewards correspond to productivity but not ability Productivity depends both on ability and on social background Nonetheless some sort of systematic positive relationship between earnings and ability might still be sought Imagine that we observe in equilibrium an economy of the type studied here Suppose that we can observe an agents income but neither his ability nor his parents income Now the pair (iux) of individual ability and income is a jointly random vector in the population A very weak definition of a meritocratic distribution of income is that iu and x are positively correlated Under this definition the distribution is meritocratic if a regression of income on ability would yield a positive coefficient in a large

855 INTERGENERATIONAL TRANSFERS

sample A more substantive restriction would require that if x gt x then pr[G gt a 1 x = x ] gt pr[G gt a I x = x2]for all a E [0 11 Under this stronger defi- nition of a meritocratic distribution of rewards we require that greater earnings imply a greater conditional probability that innate aptitude exceeds any pre- specified level The following theorem asserts that the economy studied here is generally weakly meritocratic but not strongly so

THEOREM5 Under A1-A4 the economy is weakly meritocratic However it is not generally true that the economy is strongly meritocratic

The reason that a strong meritocracy may fail to obtain can be easily seen Suppose there exists some range of parental incomes over which the investment schedule e(y) increases rapidly such that the least able offspring of parents at the top earn nearly as much as the most able offspring of parents at the bottom of this range Then the conditional ability distribution of those earning incomes slightly greater than that earned by the most able children of the least well-off parents may be dominated by the conditional distribution of ability of those whose incomes are slightly less than the earnings of the least able children of the most well-off parents An example along these lines is discussed in Section 6

5 AN EXAMPLE

It is interesting to examine the explicit solution of this problem in the special case of Cobb-Douglas production and utility functions For the analysis of this section we assume that

h ( a e ) = (ae)

f ( a ) 3 1 a E [0 1 1

With these particular functions it is not hard to see that the consistent indirect utility function and optimal investment function have the following forms

(10) V ( y ) = k y s k gt 0 0 lt 6 lt 1 and

e (y ) = sy 0 lt s lt 1

The reader may verify by direct substitution that the functions V and e given above satisfy (2)when s = ( 1 + y)2 6 = 2 y ( l + y) and

By Theorem 1 we know these solutions are unique Thus in this simple instance we obtain a constant proportional investment

function This result continues to hold for arbitrary distributions of ability f and

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

852 GLENN C LOURY

market to completely eliminate the dispersion in expected rates of return to training across families which arises in this model Legally poor parents will not be able to constrain their children to honor debts incurred on their behalf Nor will the newly matured children of wealthy families be able to attach the (human) assets of their less well-off counterparts should the latter decide for whatever reasons not to repay their loans (Default has been a pervasive problem with government guaranteed educational loan programs which would not exist absent public underwriting) Moreover the ability to make use of human capital is unknown even to the borrower Thus genuine bankruptcies can be expected as people mature to learn they are not as able as their parents gambled they would be The absence of inter-family loans in this model reflects an important feature of reality the allocative implications of which deserve study

Another critical feature of this story is that parents bear theoretically insurable risk when investing in their offspring Earnings of children are random here because the economic ability of the child becomes known only in maturity Yet a significant spreading of risk is technologically possible for a group of parents investing at the same level since their childrens earnings are independent identically distributed random variables Again we may expect that markets will fail to bring about complete risk spreading the moral hazard problems of income insurance contracts are o b v i ~ u s ~

Any attempt by a central authority to correct for these market imperfections must cope with the same incentive problems which limit the scope of the competitive mechanism here Efforts by a government to alter the allocation of training among young people will affect the consumption-investment decision of parents Moreover parents may be presumed to have better information than a central authority about the distribution of their childrens abilities (This is especially so if ability is positively correlated across generations of the same family) Yet as long as a group of parents with the same information have different incomes and there is no capital market their investment decisions will vary causing inefficiency This inefficiency could be reduced if the government were to redistribute incomes more equally but only at the cost of the excess burden accompanying such a redistributive tax

However because we have assumed altruism between generations a program of income redistribution which is imposed permanently has welfare effects here not encountered in the usual tax analysis The fact that income will be redistri- buted by taxing the next generation of workers causes each parent to regard his offsprings random income as less risky Since parents are risk averse egalitarian redistributive measures have certain welfare enhancing insurance characteris-tics It may be shown that as long as the redistributive activity is not too extensive this insurance effect dominates the excess burden effect Under such circumstances a permanent redistributive tax policy can be designed in such a way as to make all current mature agents better off

See Zeckhauser [30]for a discussion of these problems

853 INTERGENERATIONAL TRANSFERS

DEFINITION3 An education specific tax policy (ESTP) is a function r R + R+ with the interpretation that r(ye) is the after-tax income of someone whose income is y and training is e An ESTP is admissible if h(a e) = r(h(a e) e) satisfies Assumption 2 An admissible ESTP is purely redistributive (PRESTP) if j[r(h(a e) e) - h ( a e)]f(a)p (da) = 0 for every e gt 0 Denote by C the class of admissible PRESTP C is a non-empty convex set

DEFINITION4 The partial ordering is more egalitarian than denoted + is defined on C by the requirement 7 3 7 if and only if for each e the after tax distribution of income under 7 is more risky in the sense of second order stochastic dominance (see for example [24]) than the after tax distribution under

These two definitions allow us to formalize the discussion above By a result of Atkinson [I] 7 3 7 implies the Lorenze curve of the after tax income distribu- tion for persons of a given level of training under 7 lies on or above the corresponding curve under 7 However since the investment function e() will depend on 7 the same statement will not generally hold for the overall distribu- tions of income Thus we must use the term more egalitarian with some caution Let (y e) E jh(a e)f(a)p (da) and ~ ( y e) = y for all (y e) E

2+relative to Crepresenting perfect income insurance is maximal in Then on the other hand represents laisset faire For B E (0 I ) let r = B7 + (1 - )I Then r E C Finally for every r E C let V denote the unique solution for (2) when h is replaced by h

THEOREM3 Under A1-A3 7 $ r2 V(y) gt V(y) Vy gt 0 T1T2E C

Theorem 3 states that the use of a more egalitarian PRESTP among future generations of workers will lead to a higher level of welfare for each mature individual currently The careful reader will notice that this result depends upon our assumption that labor supply is exogenous In general we may expect that an income tax will distort both the educational investment decisions of parents and their choices concerning work effort By allowing the income tax function to vary with the level of educational investment (redistributing income within education classes only) we avoid the negative welfare consequences of the former distor- tion It is possible to preserve the result in Theorem 3 in the face of the latter kind of distortion as well if we restrict our attention to small redistributive efforts To see this imagine that h(ae) determines output per unit of time worked that each parent is endowed with one unit of time and that parents utility depends on work effort as well as consumption and offsprings welfare Under these slightly more general circumstances one can repeat the development of Theorems 1 and 2 with the natural modifications that indirect utility becomes a function of output per unit time (instead of total income) and income is the product of work effort and h(a e) Now consider introducing the tax scheme r (described above) for small B For fixed labor supply it may be shown that

854 GLENN C LOURY

welfare at any level of income is a concave function of B Moreover Theorem 3 implies that welfare is an increasing function of B However from standard tax theory we know that the excess burden of the tax T~ is zero when B = 0 and has derivative with respect to equal to zero at B = 0 (That is the excess burden of a small tax is a second order effect) Thus for sufficiently close to zero the insurance effect described in Theorem 3 will outweigh the excess burden effect of the tax scheme leading to an increase in wellbeing for all parents of the current generation

This result suggests that the standard tradeoff between equity and efficiency is somewhat less severe in the second-best world where opportunities to acquire training vary with parental resources It is sometimes possible to design policies which increase both equity and efficiency As a further indication of this last point let us examine the impact of public education in this model By public education we mean the centralized provision of training on an equal basis to every young person in the economy For the purposes of comparison we contrast the laissez faire outcome with that which occurs under public education with a per-capita budget equal to the expenditure of the average family in the no-intervention equilibrium We assume for this result that education is financed with non-distortionary tax

ASSUMPTION is convex in y5 h(ae(y)) is concave in y and ah(ae(y))aa

Assumption 5 posits that the marginal product of social background declines as social background (ie parents income) improves but declines less rapidly for the more able We have proven in Section 6 the following result

THEOREM4 Under A1-A5 universal public education with a per-capita budget equal to the investment of the average family in the laissez faire equilibrium will produce an earnings distribution with lower variance and higher mean than that which obtains without intervention

Finally let us turn to an examination of the relationship between earnings and ability in equilibrium It is widely held that differences in ability provide ethical grounds for differences in rewards An often cited justification for the market determined distribution of economic advantage is that it gives greater rewards to those of greater abilities In our model such is not literally the case because opportunities for training vary with social origin Thus rewards correspond to productivity but not ability Productivity depends both on ability and on social background Nonetheless some sort of systematic positive relationship between earnings and ability might still be sought Imagine that we observe in equilibrium an economy of the type studied here Suppose that we can observe an agents income but neither his ability nor his parents income Now the pair (iux) of individual ability and income is a jointly random vector in the population A very weak definition of a meritocratic distribution of income is that iu and x are positively correlated Under this definition the distribution is meritocratic if a regression of income on ability would yield a positive coefficient in a large

855 INTERGENERATIONAL TRANSFERS

sample A more substantive restriction would require that if x gt x then pr[G gt a 1 x = x ] gt pr[G gt a I x = x2]for all a E [0 11 Under this stronger defi- nition of a meritocratic distribution of rewards we require that greater earnings imply a greater conditional probability that innate aptitude exceeds any pre- specified level The following theorem asserts that the economy studied here is generally weakly meritocratic but not strongly so

THEOREM5 Under A1-A4 the economy is weakly meritocratic However it is not generally true that the economy is strongly meritocratic

The reason that a strong meritocracy may fail to obtain can be easily seen Suppose there exists some range of parental incomes over which the investment schedule e(y) increases rapidly such that the least able offspring of parents at the top earn nearly as much as the most able offspring of parents at the bottom of this range Then the conditional ability distribution of those earning incomes slightly greater than that earned by the most able children of the least well-off parents may be dominated by the conditional distribution of ability of those whose incomes are slightly less than the earnings of the least able children of the most well-off parents An example along these lines is discussed in Section 6

5 AN EXAMPLE

It is interesting to examine the explicit solution of this problem in the special case of Cobb-Douglas production and utility functions For the analysis of this section we assume that

h ( a e ) = (ae)

f ( a ) 3 1 a E [0 1 1

With these particular functions it is not hard to see that the consistent indirect utility function and optimal investment function have the following forms

(10) V ( y ) = k y s k gt 0 0 lt 6 lt 1 and

e (y ) = sy 0 lt s lt 1

The reader may verify by direct substitution that the functions V and e given above satisfy (2)when s = ( 1 + y)2 6 = 2 y ( l + y) and

By Theorem 1 we know these solutions are unique Thus in this simple instance we obtain a constant proportional investment

function This result continues to hold for arbitrary distributions of ability f and

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

853 INTERGENERATIONAL TRANSFERS

DEFINITION3 An education specific tax policy (ESTP) is a function r R + R+ with the interpretation that r(ye) is the after-tax income of someone whose income is y and training is e An ESTP is admissible if h(a e) = r(h(a e) e) satisfies Assumption 2 An admissible ESTP is purely redistributive (PRESTP) if j[r(h(a e) e) - h ( a e)]f(a)p (da) = 0 for every e gt 0 Denote by C the class of admissible PRESTP C is a non-empty convex set

DEFINITION4 The partial ordering is more egalitarian than denoted + is defined on C by the requirement 7 3 7 if and only if for each e the after tax distribution of income under 7 is more risky in the sense of second order stochastic dominance (see for example [24]) than the after tax distribution under

These two definitions allow us to formalize the discussion above By a result of Atkinson [I] 7 3 7 implies the Lorenze curve of the after tax income distribu- tion for persons of a given level of training under 7 lies on or above the corresponding curve under 7 However since the investment function e() will depend on 7 the same statement will not generally hold for the overall distribu- tions of income Thus we must use the term more egalitarian with some caution Let (y e) E jh(a e)f(a)p (da) and ~ ( y e) = y for all (y e) E

2+relative to Crepresenting perfect income insurance is maximal in Then on the other hand represents laisset faire For B E (0 I ) let r = B7 + (1 - )I Then r E C Finally for every r E C let V denote the unique solution for (2) when h is replaced by h

THEOREM3 Under A1-A3 7 $ r2 V(y) gt V(y) Vy gt 0 T1T2E C

Theorem 3 states that the use of a more egalitarian PRESTP among future generations of workers will lead to a higher level of welfare for each mature individual currently The careful reader will notice that this result depends upon our assumption that labor supply is exogenous In general we may expect that an income tax will distort both the educational investment decisions of parents and their choices concerning work effort By allowing the income tax function to vary with the level of educational investment (redistributing income within education classes only) we avoid the negative welfare consequences of the former distor- tion It is possible to preserve the result in Theorem 3 in the face of the latter kind of distortion as well if we restrict our attention to small redistributive efforts To see this imagine that h(ae) determines output per unit of time worked that each parent is endowed with one unit of time and that parents utility depends on work effort as well as consumption and offsprings welfare Under these slightly more general circumstances one can repeat the development of Theorems 1 and 2 with the natural modifications that indirect utility becomes a function of output per unit time (instead of total income) and income is the product of work effort and h(a e) Now consider introducing the tax scheme r (described above) for small B For fixed labor supply it may be shown that

854 GLENN C LOURY

welfare at any level of income is a concave function of B Moreover Theorem 3 implies that welfare is an increasing function of B However from standard tax theory we know that the excess burden of the tax T~ is zero when B = 0 and has derivative with respect to equal to zero at B = 0 (That is the excess burden of a small tax is a second order effect) Thus for sufficiently close to zero the insurance effect described in Theorem 3 will outweigh the excess burden effect of the tax scheme leading to an increase in wellbeing for all parents of the current generation

This result suggests that the standard tradeoff between equity and efficiency is somewhat less severe in the second-best world where opportunities to acquire training vary with parental resources It is sometimes possible to design policies which increase both equity and efficiency As a further indication of this last point let us examine the impact of public education in this model By public education we mean the centralized provision of training on an equal basis to every young person in the economy For the purposes of comparison we contrast the laissez faire outcome with that which occurs under public education with a per-capita budget equal to the expenditure of the average family in the no-intervention equilibrium We assume for this result that education is financed with non-distortionary tax

ASSUMPTION is convex in y5 h(ae(y)) is concave in y and ah(ae(y))aa

Assumption 5 posits that the marginal product of social background declines as social background (ie parents income) improves but declines less rapidly for the more able We have proven in Section 6 the following result

THEOREM4 Under A1-A5 universal public education with a per-capita budget equal to the investment of the average family in the laissez faire equilibrium will produce an earnings distribution with lower variance and higher mean than that which obtains without intervention

Finally let us turn to an examination of the relationship between earnings and ability in equilibrium It is widely held that differences in ability provide ethical grounds for differences in rewards An often cited justification for the market determined distribution of economic advantage is that it gives greater rewards to those of greater abilities In our model such is not literally the case because opportunities for training vary with social origin Thus rewards correspond to productivity but not ability Productivity depends both on ability and on social background Nonetheless some sort of systematic positive relationship between earnings and ability might still be sought Imagine that we observe in equilibrium an economy of the type studied here Suppose that we can observe an agents income but neither his ability nor his parents income Now the pair (iux) of individual ability and income is a jointly random vector in the population A very weak definition of a meritocratic distribution of income is that iu and x are positively correlated Under this definition the distribution is meritocratic if a regression of income on ability would yield a positive coefficient in a large

855 INTERGENERATIONAL TRANSFERS

sample A more substantive restriction would require that if x gt x then pr[G gt a 1 x = x ] gt pr[G gt a I x = x2]for all a E [0 11 Under this stronger defi- nition of a meritocratic distribution of rewards we require that greater earnings imply a greater conditional probability that innate aptitude exceeds any pre- specified level The following theorem asserts that the economy studied here is generally weakly meritocratic but not strongly so

THEOREM5 Under A1-A4 the economy is weakly meritocratic However it is not generally true that the economy is strongly meritocratic

The reason that a strong meritocracy may fail to obtain can be easily seen Suppose there exists some range of parental incomes over which the investment schedule e(y) increases rapidly such that the least able offspring of parents at the top earn nearly as much as the most able offspring of parents at the bottom of this range Then the conditional ability distribution of those earning incomes slightly greater than that earned by the most able children of the least well-off parents may be dominated by the conditional distribution of ability of those whose incomes are slightly less than the earnings of the least able children of the most well-off parents An example along these lines is discussed in Section 6

5 AN EXAMPLE

It is interesting to examine the explicit solution of this problem in the special case of Cobb-Douglas production and utility functions For the analysis of this section we assume that

h ( a e ) = (ae)

f ( a ) 3 1 a E [0 1 1

With these particular functions it is not hard to see that the consistent indirect utility function and optimal investment function have the following forms

(10) V ( y ) = k y s k gt 0 0 lt 6 lt 1 and

e (y ) = sy 0 lt s lt 1

The reader may verify by direct substitution that the functions V and e given above satisfy (2)when s = ( 1 + y)2 6 = 2 y ( l + y) and

By Theorem 1 we know these solutions are unique Thus in this simple instance we obtain a constant proportional investment

function This result continues to hold for arbitrary distributions of ability f and

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

854 GLENN C LOURY

welfare at any level of income is a concave function of B Moreover Theorem 3 implies that welfare is an increasing function of B However from standard tax theory we know that the excess burden of the tax T~ is zero when B = 0 and has derivative with respect to equal to zero at B = 0 (That is the excess burden of a small tax is a second order effect) Thus for sufficiently close to zero the insurance effect described in Theorem 3 will outweigh the excess burden effect of the tax scheme leading to an increase in wellbeing for all parents of the current generation

This result suggests that the standard tradeoff between equity and efficiency is somewhat less severe in the second-best world where opportunities to acquire training vary with parental resources It is sometimes possible to design policies which increase both equity and efficiency As a further indication of this last point let us examine the impact of public education in this model By public education we mean the centralized provision of training on an equal basis to every young person in the economy For the purposes of comparison we contrast the laissez faire outcome with that which occurs under public education with a per-capita budget equal to the expenditure of the average family in the no-intervention equilibrium We assume for this result that education is financed with non-distortionary tax

ASSUMPTION is convex in y5 h(ae(y)) is concave in y and ah(ae(y))aa

Assumption 5 posits that the marginal product of social background declines as social background (ie parents income) improves but declines less rapidly for the more able We have proven in Section 6 the following result

THEOREM4 Under A1-A5 universal public education with a per-capita budget equal to the investment of the average family in the laissez faire equilibrium will produce an earnings distribution with lower variance and higher mean than that which obtains without intervention

Finally let us turn to an examination of the relationship between earnings and ability in equilibrium It is widely held that differences in ability provide ethical grounds for differences in rewards An often cited justification for the market determined distribution of economic advantage is that it gives greater rewards to those of greater abilities In our model such is not literally the case because opportunities for training vary with social origin Thus rewards correspond to productivity but not ability Productivity depends both on ability and on social background Nonetheless some sort of systematic positive relationship between earnings and ability might still be sought Imagine that we observe in equilibrium an economy of the type studied here Suppose that we can observe an agents income but neither his ability nor his parents income Now the pair (iux) of individual ability and income is a jointly random vector in the population A very weak definition of a meritocratic distribution of income is that iu and x are positively correlated Under this definition the distribution is meritocratic if a regression of income on ability would yield a positive coefficient in a large

855 INTERGENERATIONAL TRANSFERS

sample A more substantive restriction would require that if x gt x then pr[G gt a 1 x = x ] gt pr[G gt a I x = x2]for all a E [0 11 Under this stronger defi- nition of a meritocratic distribution of rewards we require that greater earnings imply a greater conditional probability that innate aptitude exceeds any pre- specified level The following theorem asserts that the economy studied here is generally weakly meritocratic but not strongly so

THEOREM5 Under A1-A4 the economy is weakly meritocratic However it is not generally true that the economy is strongly meritocratic

The reason that a strong meritocracy may fail to obtain can be easily seen Suppose there exists some range of parental incomes over which the investment schedule e(y) increases rapidly such that the least able offspring of parents at the top earn nearly as much as the most able offspring of parents at the bottom of this range Then the conditional ability distribution of those earning incomes slightly greater than that earned by the most able children of the least well-off parents may be dominated by the conditional distribution of ability of those whose incomes are slightly less than the earnings of the least able children of the most well-off parents An example along these lines is discussed in Section 6

5 AN EXAMPLE

It is interesting to examine the explicit solution of this problem in the special case of Cobb-Douglas production and utility functions For the analysis of this section we assume that

h ( a e ) = (ae)

f ( a ) 3 1 a E [0 1 1

With these particular functions it is not hard to see that the consistent indirect utility function and optimal investment function have the following forms

(10) V ( y ) = k y s k gt 0 0 lt 6 lt 1 and

e (y ) = sy 0 lt s lt 1

The reader may verify by direct substitution that the functions V and e given above satisfy (2)when s = ( 1 + y)2 6 = 2 y ( l + y) and

By Theorem 1 we know these solutions are unique Thus in this simple instance we obtain a constant proportional investment

function This result continues to hold for arbitrary distributions of ability f and

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

855 INTERGENERATIONAL TRANSFERS

sample A more substantive restriction would require that if x gt x then pr[G gt a 1 x = x ] gt pr[G gt a I x = x2]for all a E [0 11 Under this stronger defi- nition of a meritocratic distribution of rewards we require that greater earnings imply a greater conditional probability that innate aptitude exceeds any pre- specified level The following theorem asserts that the economy studied here is generally weakly meritocratic but not strongly so

THEOREM5 Under A1-A4 the economy is weakly meritocratic However it is not generally true that the economy is strongly meritocratic

The reason that a strong meritocracy may fail to obtain can be easily seen Suppose there exists some range of parental incomes over which the investment schedule e(y) increases rapidly such that the least able offspring of parents at the top earn nearly as much as the most able offspring of parents at the bottom of this range Then the conditional ability distribution of those earning incomes slightly greater than that earned by the most able children of the least well-off parents may be dominated by the conditional distribution of ability of those whose incomes are slightly less than the earnings of the least able children of the most well-off parents An example along these lines is discussed in Section 6

5 AN EXAMPLE

It is interesting to examine the explicit solution of this problem in the special case of Cobb-Douglas production and utility functions For the analysis of this section we assume that

h ( a e ) = (ae)

f ( a ) 3 1 a E [0 1 1

With these particular functions it is not hard to see that the consistent indirect utility function and optimal investment function have the following forms

(10) V ( y ) = k y s k gt 0 0 lt 6 lt 1 and

e (y ) = sy 0 lt s lt 1

The reader may verify by direct substitution that the functions V and e given above satisfy (2)when s = ( 1 + y)2 6 = 2 y ( l + y) and

By Theorem 1 we know these solutions are unique Thus in this simple instance we obtain a constant proportional investment

function This result continues to hold for arbitrary distributions of ability f and

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

856 GLENN C LOURY

for production functions h(a e) = aae- a E (0l) However only in the case given in (9) are we able to explicitly derive the equilibrium distribution the task to which we now turn

A parent earning y invests sy in his offspring whose income is then x = us^)^ Notice that y gt s =+x lt y no matter what a turns out to be Thus no familys income can remain indefinitely above s and the equilibrium distribution is concentrated on [Os] Now h- (x e(y)) = x2sy y E [x2ss] gives the ability necessary to reach earnings x when parental income is y Let gn() be the density function of the income distribution in period n Then gn(x)dx is for small dx the fraction of families in period n with incomes in the interval (x x + dx) Heuristically

=C f ( h l ( s y ) ) ah -

( s ~ ) g n - l ( ~ ) d x Y

This suggests the following formula shown rigorously to be valid in Lemma 1 of Section 6

Thus we have that the density of the equilibrium distribution solves the func- tional equation

Consider now the change of variables z = xs Then z varies in [Ol] and the density of z satisfies g(z) = si(sz) One may now use (1 1) (with a change of variables y = y s in the integrand) to see that g() must satisfy

A solution of (12) gives immediately the solutions of (11) for all values of s E (0l) Theorem 2 assures the existence and uniqueness of such a solution

We solve (12) by converting it into a functional differential equation presum- ing its solution may be written in series form and then finding the coefficients for this series Differentiating (12) we have that

-dg (z) = g(z)z - 4f(z2)dz

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

857 INTERGENERATIONAL TRANSFERS

Suppose that g(z) = C==znSubstitute this form into (13) Equate coeffi- cients of like powers on the left and right hand sides and deduce the following recursive system for a)

a = g arbitrary and

Solving directly for a ) yields

Thus we have that the unique solution of (12) is given by

The constant g is determined by the requirement that g integrates to unity ( g =692)

The implied solution for g(x) may be derived from this formula for any s E (0 I) The basic character of the equilibrium distribution is unaffected by the savings rate (ie the parameter y in the utility function) which simply alters its scale The equilibrium distributions are single peaked just slightly right skewed with a mean of (0422)s and a variance of (0037)s2 Consumption per family is maximized in equilibrium when s = 12 the output elasticity of education Public education in this example with a per capita budget equal to the training expenditure of the average family in equilibrium reduces the variance of the equilibrium distribution by 35 per cent to (0024)s2 At the same time mean output rises by 34 per cent under the centralized provision of training resources In this example the efficiency gain of centralized training is small relative to the associated reduction in inequality

PROOFOF THEOREM1 Recall the map T C [0 J ]-+ C[0 J ] defined by

(T)(Y)= oyampEa U(c (h(a y - c)))

For E C[O J ] define 1 1 ( 1 We shall show that under this - s~p [~ ~~((x) l

858 GLENN C LOURY

norm T is a contraction on C[0 j ] Let B I E C[0 J ]

1 1 T - I = sup I man E U(c(h(a y - c)) y gt y gt o O G C G Y

- max EU(cI(h(a y - c ) ) ) O s c G y

Let t (y ) give the maximum for EU(cB) and $() give the maximum for E U(c I) Then

1 T B - TI11 = sup I E(U(E B) - u(amp I))] p gt y a O

lt sup max 1 B) - U(cI)) I E ( ~ ( c y gt y gt o

lt m a n ( sup E(U2(t)[B-])I g gt y gt o

sup j(h(a y - c(y)))- (h(ay - c ( ~ ) ) ) j g gt y gt o UE[O 11

for some y lt 1 using Al(iii) Hence T is a contraction and by the Banach Fixed Point Theorem [ll Theorem 382 p 1191 there exists a unique fixed point V E C [0 j ] By the definition of T V is the solution of (2)

Define the sequence of functions Vn) E C[Oj ] inductively as follows

Clearly V) + V uniformly Let t n ( y ) be the optimal policy function corre- sponding to V It follows from Al(i) A2(i) the Implicit Function Theorem and an induction that t n ( y ) are differentiable functions n = 12 Moreover Al(ii) A2(i) A3 the Envelope Theorem and another induction imply that V is differentiable and limysdVndy(y) = oon = 12 Thus 0 lt t n ( y ) lty Vn and Vy E (0 j ]

859 INTERGENERATIONAL TRANSFERS

Now it follows from the strict concavity of U the concavity of h in its second argument and an induction that the functions Vn are strictly concave Hence V is concave But for 0 lt S lt 1 using the above-mentioned concavity we have

v(syl + (1 - s)y2)= maxEU(c V(h(aSy + (1 - S)y2- c)))

where c - c(y) and c = c(y) with equality iffy =y Hence V is strictly concave and therefore differentiable almost everywhere Now the Envelope Theorem implies

dVt ( y ) = E(c V) E ( h ) + h ( a y - c) d~ dy

while

dv--(Y)d~ = d~ - c))~ ( u ( c v) c ( h ) h2(a y

where + and - refer to right or left hand derivatives respectively It follows from the monotonicity of h in a the continuity off and the fact that dV+dy dV-dy at most on a set of y-measure zero that the right-hand side of the two equations above are equal Hence V is differentiable on (0 j ] and (by uniform convergence of Vn) limodVdy(y) = co Thus c(y) is unique for each y and 0 lt c(y) lty Notice now that t n ) c pointwise on [O 81 If this were not true then for some y the sequence t n ( y ) ) would possess a convergent subsequence bounded away from c(y) contradicting the unique- ness of the latter The continuity of c follows

PROOFOF THEOREM2 We want to show that the Markoff chain

possesses a unique globally stable invariant measure v E 9 where the transi- tion probability is given by

The mathematical question here is precisely that which arises in the theory of stochastic growth (see for example Brock-Mirman [S]) To answer this question

860 GLENN C LOURY

we use the following results reported in the excellent recent survey of Futia

PROPOSITION(Futia [12 Theorems 36 37 46 52 and 561) Under Assump- tions A1-A3 the process represented by equations (5) and (6) has at least one invariant measure v E9and aIways converges to some invariant measure More- over if in addition A4 and the foIIowing condition hold then the invariant measure is unique

CONDITION1 There exists a yo E [O 81 with the property that for every integer k gt 1 any number y E [0 y] and any neighborhood U of yo one can find an integer n such that P ~ ( ~ U ) gt 0

The m-step transition probability P m ( y A ) is defined inductively by P l ( y ~ ) r P ( y A ) and

Thus we will have proven the theorem if we can exhibit an earnings level yo every neighborhood of which is with positive probability and finite periodicity entered infinitely often from any initial earnings level We will need the following notation

The following Lemmata will also be useful

PROOF Implement the change of variables a = h - ( x e(y)) in (6) QED

REMARKThus our transition probability on the bounded state space [0 y] arises from integrating a bounded measurable density function It is this fact which enables us to employ Futias results indicated in the Proposition above

LEMMA2 Let S m ( y ) be the support of the probabiIity P m ( y ) E9Then S m ( y ) = (x(y)x(y))

PROOF Define

= 0 otherwise

INTERGENERATIONAL TRANSFERS

and inductively

Now by Lemma 1 we have

Thus Sm(y) = x I pm(y x )gt O) and we need only show that pe(yx) gt 0 iff x E (xr (y) x(y)) This is done by induction Obviously pl(x y) gt 0 iff x E (x(~) xi(y)) from the definitions Suppose the hypothesis holds for pm- (Y x) and define Am(x y) -- z E [0 co) ( P ( ~ Z )~ - (Z x) gt 0) Now pn(y x) gt 0 iff p(Am(x y)) gt 0 ~ u t gt 0 iff z E (X(Y)X(Y)) whilepm-(zx) P ( ~ Z ) gt 0 iff x E (x-(t)x-(z)) It follows from Assumptions A2(i) and A4 that the functions x(-) and x() are strictly increasing Thus

So p(Am(x y)) gt 0 iff both (x-I)-(x) lt and (x-I)-(x) gt xi(y) that is iff x(y) gt x gt x(y) QED

LEMMA3 Lim xom(y)= 0 Vy E [0 71 Moreover there exists y gt 0 such that lim inf x(y) gt y Vy E [0 J]

PROOF For y gt 0 by A2(i) and Theorem 1 xi(y) lt e(y) lty Thus the sequence of numbers x(y)) is monotonically decreasing and bounded below by zero Let x = lirn x(y) Then if g gt0 we have by continuity of xi(-)

a contradiction The second statement in the lemma follows from the fact that x(O) gt 0 (by A2(i)) so by continuity there exists y gt 0 such that x ] ( ~ ) gty Vy lt 9 Thus x(y)) could never have a convergent subsequence approaching a value less than y QED

We can now complete the argument for Condition 1 By Lemma 2 Pm(yA) gt 0 if p(A n (x(y) x(y))) gt 0 By Lemma 3 there is an interval (O) such that [A C (0 y)] [ A c (x(y) x(y))] with y given and m sufficiently large Thus for any point yo E (0 y) we have shown that every neighborhood of yo has positive probability of being entered after a finite number of steps from any original pointy E [0 oo) Moreover we have shown that Pm(yA) gt Opmk(y A) gt 0 for k gt 1 and A c (0 y) Condition 1 then follows

It remains to argue that the support of the invariant measure is connected and compact But this follows from A2(iii) and A4 which imply the exi3tence of a smallest point F E [0 oo) with the property that x ] ( ~ ) lty Vy gtF Thus when

862 GLENN C LOURY

y gtJthe supremum of the support of Pm(y ) monotonically decreases until it falls below JHence the invariant measure has support contained in the interval [Oy=] It is clear that the supremum of the support of the invariant measure y satisfies y = Moreover limmx(y) 0 V y implies that if y is in the ~ ( ~ 3 =

support of the invariant measure and y lty then y is in the support Hence the support is connected QED

PROOFOF THEOREM3 For any t E12the map T C [0y]+C [0y] is defined by

T ( y )= max E U(c ( t (h (a y - c) y - c)) ) y E [ O J]o lt c lt y

Inductively define T= T ( T - I ) n = 23 By Theorem 1 we know there is a unique y E C[Oy] such that lim I(T)(y)- V(y)( = 0 V y E [0J] Now let ( y )= U(yO)and suppose t + t Then we have

(T )(Y)= oppyEm U ( t ( h ( f f Y - c) Y - c)O))

lt max E U(c ~ ( 1 ( h ( a - c) y - c) 0 ) ) o lt c lt y

by thf facts that the composition of increasing concave functions is concave and that t + t implies the expectation of any concave function of after tax income is no less under t than under t Suppose inductively that ( T - ) ( ~ ) lt ( T - I )

( y ) V y E LO 71 Then

= max (T- )(t(h(a - c) y - c ) ) )E ~ ( c o lt c lt y

4 max E U ( C ( T P 1 ) ( t ( h ( a - c) y - c ) ) )o lt c lt y

lt man (T ) ( i ( h ( a - c) y - c ) ) )E ~ ( c o lt c lt y

where the first inequality follows from the induction hypothesis and monotonic- ity of U while the second inequality is established by the argument employed just above Hence we have shown that (T)(y)4 (Tp)(y )by gt 0 V n = 1 2 It then follows from pointwise convergence that V(y) lt VP(y) V y E

[O JI QED

PROOFOF THEOREM4 Let 7 be the average income in the laissez faire equilibrium Denote by x the earnings of an arbitrary mature individual and by

863 INTERGENERATIONAL TRANSFERS

y the earnings of his parent in the preceding period In what follows we will think of (x y) as a jointly random vector Now it is a well known fact of bivariate distribution theory that

Moreover the variance of income under public education is given by

var = var(x ( y = x)

The proof proceeds by noting that A5 implies var(x I y) is a convex function of y One may then apply Jensens inequality which along with (14) implies the result

Consider

Thus the conditional variance of offsprings earnings increases (decreases) with parental income iff ability and education are complements (substitutes) Further- more it follows from (14) that var(x) gt var(x ( y = x) when a2haa ae = 0 Now one may calculate

This term is nonnegative under A5 It follows that var(x) gt var Moreover the concavity of h(a e(y)) in y also implies x lt jflf(a)h(a e(x))p (da) so mean income is no lower under public education QED

864 GLENN C LOURY

PROOFOF THEOREM5 Fubinis Theorem implies that in equilibrium

Now v invariant implies

l o I I f ( a ) [ j 0[O 00) (h(a e(y)) - y ) v ( d y ) ] y(da) =

which simply states that per capita earnings is constant across a generation By A2(i) the bracketed term in the integrand above is strictly increasing in a Thus 36 E (0l) such that

f (a )J ( h ( a e(y)) -y)v (dy) 20 as a 1 6 10 a)

Hence

We now outline an argument showing that the economy neednt be strongly meritocratic Consider the following

where ( 2 3 is the random vector of offspring-parent earnings and y- (a x) is defined by h ( a e ( y - ( a ~ ) ) )r x That is an individual observed to have earnings x will have an innate endowment exceeding a if and only if that individuals parent had an income less than yP (a x) Let p (y x ) be as defined in the proof of Lemma 2 under the proof of Theorem 2 given above Then we have that

Thus the equilibrium distribution is meritocratic in the strong sense if and only if the right-hand side above is an increasing function of x Notice that this

865 INTERGENERATIONAL TRANSFERS

restriction is in principle testable when the distribution of parental income conditional on offsprings income can be observed

An increase in x has two effects First it leads to an increase in y- (a x) the critical level below which parental income must lie if 6 is to exceed a However an increase in x also shifts to the right the conditional distribution of parents earnings (Ylx) tending to reduce the right-hand side above That this latter effect can outweigh the former may be seen by the following (admittedly extreme) example Suppose there are only two possible levels of training invest- ment which parents can make q and 2 with q lt 2 Imagine further that -x = h(g 1) lt h(P 0) = X Now imagine observing two individuals one of whom earns 2 + r the other of whom earns x - E where E is a small positive number Then it must be the case (by continuity of h( )) that the person with earnings X + E has ability close to zero while the person who earns x- r has ability close to one QED

7 CONCLUSIONS

The analysis presented here brings together two separate strands of the literature on economic inequality On the one hand we have modelled the income dynamics as a stochastic process and studied the properties of its ergodic distribution This approach has a long history (for example [6 7 19 25 281) On the other hand the underlying structure of this stochastic process results from maximizing decisions of the economic agents As such it has much in common with the traditional human capital approach to earnings [2 3 20 211 Yet we have departed from the human capital school in one crucial respect-the assumption that a perfect market for educational loans exists We have assumed instead a balkanized market where each family must generate internally funds for the training of its young We believe this assumption parallels more closely actual circumstances As a consequence of this assumption family background exercises an independent constraining effect on social mobility Our analysis is in this regard consistent with the approach to the study of mobility which economists and sociologists have found so fruitful in recent years (for example [4 8 10 151)

The present approach commends itself in one important additional respect By grounding training decisions in rational choice we are able to rigorously analyze some normative issues regarding public policies which would alter the distribu- tion of income Moreover this analysis [Section 41 has not presupposed the existence of some social welfare function Rather it has proceeded on a purely individualistic basis We have seen that redistributive policies can serve to improve both the allocation of risks and of training resources when capital markets are incomplete Indeed normative prescriptions similar in spirit to those of Harsanyi [14] or Rawls [23] to the extent that they are derived from an analysis of the risks which an individual faces in some ex ante original position are here given a positive behavioral grounding We have replaced the

866 GLENN C LOURY

veil of ignorance with the generation gap across which parents cannot look to perceive the position which their offspring will occupy in society

Finally we should note some shortcomings of the foregoing analysis By far the strongest assumption we have made is that innate abilities are independent across generations While the exact nature of intergenerational correlations has not yet been satisfactorily ascertained the existence of some such effects is impossible to deny Theorems 1 and 2 survive with slight modification general- ization to a Markovian structure on abilities However the tax analysis of Section 4 is lost once such allowance is made This seems a stimulating topic for further investigation

Another weak point is the fact that the welfare implications of income taxation studied in Section 4 are limited to education-specific tax systems These may not be politically or administratively feasible and do not involve redistribution across social classes In any case a similar analysis of tax systems dependent only upon income would be desirable This appears to be a very difficult matter in general Moreover a strong result like that given in Theorem 3 will no longer hold for such schemes

Finally the framework developed here suggests an interesting research prob- lem in the normative economics of social mobility The transition probability for a society P ( y A ) contains all relevant information regarding both cross-section inequality in the long run (if the process is ergodic) as well as intergenerational mobility One might consider axiom systems to impose on orderings of such transition functions which reflect intuitive notions of what it means to say that one social structure is more equal than another The question then becomes when can orderings satisfying these intuitive axioms be represented in the simple way suggested in equation (S) (That is when may social structures P be ranked by the expected value of some indirect utility function under their ergodic distribution)

Universizy of Michigan

Manuscript received July 1978 revision received June 1980

I3see Goldberger [13] for an informative discussion of the problems here

REFERENCES

[I] ATKINSONA On the Measurement of Inequality Journal of Economic Theory 2 (1970) 244-263

[2] BECKERG Human Capital 2nd Edition New York National Bureau of Economic Research 1975

(31 BECKER G AND N TOMES An Equilibrium Theory of the Distribution of Income and Intergenerational Mobility Journal of Political Economy 87 (1979) 1153-1 189

[4] BOUDONR Mathematical Structures of Social Mobiliiy Amsterdam Elsevier Scientific Publish- ing Co 1973

[S] BROCKW AND L MIRMANOptimal Economic Growth and Uncertainty The Discounted Case Journal of Economic Theory 4 (1972) 479-513

INTERGENERATIONAL TRANSFERS 867

[6] CHAMPERNOWNE Model of Economic Journal 63 (1953) D G A Income Distribution 318-351

[7] CONLISKJ Can Equalization of Opportunity Reduce Social Mobility American Economic Review 64 (1974) 80-90

[S] DATCHERL The Effect of Family and Community Background on the Education and Earnings of Black and White Men Unpublished Manuscript Institute for Social Research University of Michigan 1980

[9] DIAMONDP National Debt in a Neoclassical Growth Model American Economic Review 55 (1965) 1126-1 150

[lo] DUNCAN O D FEATHERMAN AND B DUNCAN S O C ~ O ~ C O ~ O ~ ~ CBackground and Achievement New York Seminar Press 1972

[ l l ] FRIEDMANA Foundations of Modern Analysis New York Holt Rinehart and Winston Inc 1970

[12] FuTIA C A Stochastic Approach to Economic Dynamics Bell Telephone Laboratories Discussion Paper 1976

[13] GOLDBERGERA S Models and Methods in the IQ Debate Part I SSRI Workshop Series No 7710 University of Wisconsin 1977

[14] HARSANYIJ C Cardinal Welfare Individualistic Ethics and Interpersonal Comparisons of Utility Journal of Political Economy 63 (1955) 309-321

[IS] JENCKSC ET ALInequality A Reassessment of the Effect of Family and Schooling in America New York Harper and Row 1972

[16] KOHLBERGE A Model of Economic Growth with Altruism between Generations Journal of Economic Theory 13 (1976) 1-13

[17] KOOPMANST Representation of Preference Orderings over Time in Decisions and Organiza- tion ed by C B McGuire and R Radner Amsterdam North-Holland Publishing Co 1972

[IS] LOURYG C Essays in the Theory of the Distribution of Income PhD Thesis Massachu- setts Institute of Technology 1976

[19] MANDELBROTB Stable Paretian Random Functions and the Multiplicative Variation of Income Econometrica 29 (1961) 517-543

[20] MINCERJ Investment in Human Capital and Personal Income Distribution Journal of Political Economy 66 (1958) 281-302

[211 -- The Distribution of Labor Incomes A Survey Journal of Economic Literature 8 (1970) 1-26

[22] OKUNAEquity and Eficiency The Big Tradeoff Washington D C The Brookings Institu- tion 1975

1231 RAWL~J A Theory of Justice Cambridge Massachusetts Harvard University Press 1971 24 ROTHSCHILD M AND J STIGLITZ Increasing Risk I A Definition Journal of Economic

Theory 2 (1970) 225-243 [25] ROY A D The Distribution of Earnings and Individual Output Economic Journal 60

(1950) 489-505 [26] SAMUELSON An Exact Consumption-Loan Model of Interest with or without the Social P

Contrivance of Money Journal of Political Economy 66 (1958) 467-482 [27] SHORROCKSA The Measurement of Mobility Econometrica 46 (1978) 1013-1024 [28] SOLOW R On the Dynamics of the Income Distribution PhD Thesis Harvard University

1951 1291 VARIAN H Redistributive Taxation as Social Insurance Journal o f Public Economics 14

(1980) 49-68 [30] ZECKHAUSERR Risk Spreading and Distribution Chapter 8 in Redistribution Through Public

Choice ed by Hochman and Peterson New York Columbia University Press 1974

You have printed the following article

Intergenerational Transfers and the Distribution of EarningsGlenn C LouryEconometrica Vol 49 No 4 (Jul 1981) pp 843-867Stable URL

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[Footnotes]

4 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

9 National Debt in a Neoclassical Growth ModelPeter A DiamondThe American Economic Review Vol 55 No 5 Part 1 (Dec 1965) pp 1126-1150Stable URL

httplinksjstororgsicisici=0002-82822819651229553A53C11263ANDIANG3E20CO3B2-L

References

3 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

httpwwwjstororg

LINKED CITATIONS- Page 1 of 3 -

NOTE The reference numbering from the original has been maintained in this citation list

859 INTERGENERATIONAL TRANSFERS

Now it follows from the strict concavity of U the concavity of h in its second argument and an induction that the functions Vn are strictly concave Hence V is concave But for 0 lt S lt 1 using the above-mentioned concavity we have

v(syl + (1 - s)y2)= maxEU(c V(h(aSy + (1 - S)y2- c)))

where c - c(y) and c = c(y) with equality iffy =y Hence V is strictly concave and therefore differentiable almost everywhere Now the Envelope Theorem implies

dVt ( y ) = E(c V) E ( h ) + h ( a y - c) d~ dy

while

dv--(Y)d~ = d~ - c))~ ( u ( c v) c ( h ) h2(a y

where + and - refer to right or left hand derivatives respectively It follows from the monotonicity of h in a the continuity off and the fact that dV+dy dV-dy at most on a set of y-measure zero that the right-hand side of the two equations above are equal Hence V is differentiable on (0 j ] and (by uniform convergence of Vn) limodVdy(y) = co Thus c(y) is unique for each y and 0 lt c(y) lty Notice now that t n ) c pointwise on [O 81 If this were not true then for some y the sequence t n ( y ) ) would possess a convergent subsequence bounded away from c(y) contradicting the unique- ness of the latter The continuity of c follows

PROOFOF THEOREM2 We want to show that the Markoff chain

possesses a unique globally stable invariant measure v E 9 where the transi- tion probability is given by

The mathematical question here is precisely that which arises in the theory of stochastic growth (see for example Brock-Mirman [S]) To answer this question

860 GLENN C LOURY

we use the following results reported in the excellent recent survey of Futia

PROPOSITION(Futia [12 Theorems 36 37 46 52 and 561) Under Assump- tions A1-A3 the process represented by equations (5) and (6) has at least one invariant measure v E9and aIways converges to some invariant measure More- over if in addition A4 and the foIIowing condition hold then the invariant measure is unique

CONDITION1 There exists a yo E [O 81 with the property that for every integer k gt 1 any number y E [0 y] and any neighborhood U of yo one can find an integer n such that P ~ ( ~ U ) gt 0

The m-step transition probability P m ( y A ) is defined inductively by P l ( y ~ ) r P ( y A ) and

Thus we will have proven the theorem if we can exhibit an earnings level yo every neighborhood of which is with positive probability and finite periodicity entered infinitely often from any initial earnings level We will need the following notation

The following Lemmata will also be useful

PROOF Implement the change of variables a = h - ( x e(y)) in (6) QED

REMARKThus our transition probability on the bounded state space [0 y] arises from integrating a bounded measurable density function It is this fact which enables us to employ Futias results indicated in the Proposition above

LEMMA2 Let S m ( y ) be the support of the probabiIity P m ( y ) E9Then S m ( y ) = (x(y)x(y))

PROOF Define

= 0 otherwise

INTERGENERATIONAL TRANSFERS

and inductively

Now by Lemma 1 we have

Thus Sm(y) = x I pm(y x )gt O) and we need only show that pe(yx) gt 0 iff x E (xr (y) x(y)) This is done by induction Obviously pl(x y) gt 0 iff x E (x(~) xi(y)) from the definitions Suppose the hypothesis holds for pm- (Y x) and define Am(x y) -- z E [0 co) ( P ( ~ Z )~ - (Z x) gt 0) Now pn(y x) gt 0 iff p(Am(x y)) gt 0 ~ u t gt 0 iff z E (X(Y)X(Y)) whilepm-(zx) P ( ~ Z ) gt 0 iff x E (x-(t)x-(z)) It follows from Assumptions A2(i) and A4 that the functions x(-) and x() are strictly increasing Thus

So p(Am(x y)) gt 0 iff both (x-I)-(x) lt and (x-I)-(x) gt xi(y) that is iff x(y) gt x gt x(y) QED

LEMMA3 Lim xom(y)= 0 Vy E [0 71 Moreover there exists y gt 0 such that lim inf x(y) gt y Vy E [0 J]

PROOF For y gt 0 by A2(i) and Theorem 1 xi(y) lt e(y) lty Thus the sequence of numbers x(y)) is monotonically decreasing and bounded below by zero Let x = lirn x(y) Then if g gt0 we have by continuity of xi(-)

a contradiction The second statement in the lemma follows from the fact that x(O) gt 0 (by A2(i)) so by continuity there exists y gt 0 such that x ] ( ~ ) gty Vy lt 9 Thus x(y)) could never have a convergent subsequence approaching a value less than y QED

We can now complete the argument for Condition 1 By Lemma 2 Pm(yA) gt 0 if p(A n (x(y) x(y))) gt 0 By Lemma 3 there is an interval (O) such that [A C (0 y)] [ A c (x(y) x(y))] with y given and m sufficiently large Thus for any point yo E (0 y) we have shown that every neighborhood of yo has positive probability of being entered after a finite number of steps from any original pointy E [0 oo) Moreover we have shown that Pm(yA) gt Opmk(y A) gt 0 for k gt 1 and A c (0 y) Condition 1 then follows

It remains to argue that the support of the invariant measure is connected and compact But this follows from A2(iii) and A4 which imply the exi3tence of a smallest point F E [0 oo) with the property that x ] ( ~ ) lty Vy gtF Thus when

862 GLENN C LOURY

y gtJthe supremum of the support of Pm(y ) monotonically decreases until it falls below JHence the invariant measure has support contained in the interval [Oy=] It is clear that the supremum of the support of the invariant measure y satisfies y = Moreover limmx(y) 0 V y implies that if y is in the ~ ( ~ 3 =

support of the invariant measure and y lty then y is in the support Hence the support is connected QED

PROOFOF THEOREM3 For any t E12the map T C [0y]+C [0y] is defined by

T ( y )= max E U(c ( t (h (a y - c) y - c)) ) y E [ O J]o lt c lt y

Inductively define T= T ( T - I ) n = 23 By Theorem 1 we know there is a unique y E C[Oy] such that lim I(T)(y)- V(y)( = 0 V y E [0J] Now let ( y )= U(yO)and suppose t + t Then we have

(T )(Y)= oppyEm U ( t ( h ( f f Y - c) Y - c)O))

lt max E U(c ~ ( 1 ( h ( a - c) y - c) 0 ) ) o lt c lt y

by thf facts that the composition of increasing concave functions is concave and that t + t implies the expectation of any concave function of after tax income is no less under t than under t Suppose inductively that ( T - ) ( ~ ) lt ( T - I )

( y ) V y E LO 71 Then

= max (T- )(t(h(a - c) y - c ) ) )E ~ ( c o lt c lt y

4 max E U ( C ( T P 1 ) ( t ( h ( a - c) y - c ) ) )o lt c lt y

lt man (T ) ( i ( h ( a - c) y - c ) ) )E ~ ( c o lt c lt y

where the first inequality follows from the induction hypothesis and monotonic- ity of U while the second inequality is established by the argument employed just above Hence we have shown that (T)(y)4 (Tp)(y )by gt 0 V n = 1 2 It then follows from pointwise convergence that V(y) lt VP(y) V y E

[O JI QED

PROOFOF THEOREM4 Let 7 be the average income in the laissez faire equilibrium Denote by x the earnings of an arbitrary mature individual and by

863 INTERGENERATIONAL TRANSFERS

y the earnings of his parent in the preceding period In what follows we will think of (x y) as a jointly random vector Now it is a well known fact of bivariate distribution theory that

Moreover the variance of income under public education is given by

var = var(x ( y = x)

The proof proceeds by noting that A5 implies var(x I y) is a convex function of y One may then apply Jensens inequality which along with (14) implies the result

Consider

Thus the conditional variance of offsprings earnings increases (decreases) with parental income iff ability and education are complements (substitutes) Further- more it follows from (14) that var(x) gt var(x ( y = x) when a2haa ae = 0 Now one may calculate

This term is nonnegative under A5 It follows that var(x) gt var Moreover the concavity of h(a e(y)) in y also implies x lt jflf(a)h(a e(x))p (da) so mean income is no lower under public education QED

864 GLENN C LOURY

PROOFOF THEOREM5 Fubinis Theorem implies that in equilibrium

Now v invariant implies

l o I I f ( a ) [ j 0[O 00) (h(a e(y)) - y ) v ( d y ) ] y(da) =

which simply states that per capita earnings is constant across a generation By A2(i) the bracketed term in the integrand above is strictly increasing in a Thus 36 E (0l) such that

f (a )J ( h ( a e(y)) -y)v (dy) 20 as a 1 6 10 a)

Hence

We now outline an argument showing that the economy neednt be strongly meritocratic Consider the following

where ( 2 3 is the random vector of offspring-parent earnings and y- (a x) is defined by h ( a e ( y - ( a ~ ) ) )r x That is an individual observed to have earnings x will have an innate endowment exceeding a if and only if that individuals parent had an income less than yP (a x) Let p (y x ) be as defined in the proof of Lemma 2 under the proof of Theorem 2 given above Then we have that

Thus the equilibrium distribution is meritocratic in the strong sense if and only if the right-hand side above is an increasing function of x Notice that this

865 INTERGENERATIONAL TRANSFERS

restriction is in principle testable when the distribution of parental income conditional on offsprings income can be observed

An increase in x has two effects First it leads to an increase in y- (a x) the critical level below which parental income must lie if 6 is to exceed a However an increase in x also shifts to the right the conditional distribution of parents earnings (Ylx) tending to reduce the right-hand side above That this latter effect can outweigh the former may be seen by the following (admittedly extreme) example Suppose there are only two possible levels of training invest- ment which parents can make q and 2 with q lt 2 Imagine further that -x = h(g 1) lt h(P 0) = X Now imagine observing two individuals one of whom earns 2 + r the other of whom earns x - E where E is a small positive number Then it must be the case (by continuity of h( )) that the person with earnings X + E has ability close to zero while the person who earns x- r has ability close to one QED

7 CONCLUSIONS

The analysis presented here brings together two separate strands of the literature on economic inequality On the one hand we have modelled the income dynamics as a stochastic process and studied the properties of its ergodic distribution This approach has a long history (for example [6 7 19 25 281) On the other hand the underlying structure of this stochastic process results from maximizing decisions of the economic agents As such it has much in common with the traditional human capital approach to earnings [2 3 20 211 Yet we have departed from the human capital school in one crucial respect-the assumption that a perfect market for educational loans exists We have assumed instead a balkanized market where each family must generate internally funds for the training of its young We believe this assumption parallels more closely actual circumstances As a consequence of this assumption family background exercises an independent constraining effect on social mobility Our analysis is in this regard consistent with the approach to the study of mobility which economists and sociologists have found so fruitful in recent years (for example [4 8 10 151)

The present approach commends itself in one important additional respect By grounding training decisions in rational choice we are able to rigorously analyze some normative issues regarding public policies which would alter the distribu- tion of income Moreover this analysis [Section 41 has not presupposed the existence of some social welfare function Rather it has proceeded on a purely individualistic basis We have seen that redistributive policies can serve to improve both the allocation of risks and of training resources when capital markets are incomplete Indeed normative prescriptions similar in spirit to those of Harsanyi [14] or Rawls [23] to the extent that they are derived from an analysis of the risks which an individual faces in some ex ante original position are here given a positive behavioral grounding We have replaced the

866 GLENN C LOURY

veil of ignorance with the generation gap across which parents cannot look to perceive the position which their offspring will occupy in society

Finally we should note some shortcomings of the foregoing analysis By far the strongest assumption we have made is that innate abilities are independent across generations While the exact nature of intergenerational correlations has not yet been satisfactorily ascertained the existence of some such effects is impossible to deny Theorems 1 and 2 survive with slight modification general- ization to a Markovian structure on abilities However the tax analysis of Section 4 is lost once such allowance is made This seems a stimulating topic for further investigation

Another weak point is the fact that the welfare implications of income taxation studied in Section 4 are limited to education-specific tax systems These may not be politically or administratively feasible and do not involve redistribution across social classes In any case a similar analysis of tax systems dependent only upon income would be desirable This appears to be a very difficult matter in general Moreover a strong result like that given in Theorem 3 will no longer hold for such schemes

Finally the framework developed here suggests an interesting research prob- lem in the normative economics of social mobility The transition probability for a society P ( y A ) contains all relevant information regarding both cross-section inequality in the long run (if the process is ergodic) as well as intergenerational mobility One might consider axiom systems to impose on orderings of such transition functions which reflect intuitive notions of what it means to say that one social structure is more equal than another The question then becomes when can orderings satisfying these intuitive axioms be represented in the simple way suggested in equation (S) (That is when may social structures P be ranked by the expected value of some indirect utility function under their ergodic distribution)

Universizy of Michigan

Manuscript received July 1978 revision received June 1980

I3see Goldberger [13] for an informative discussion of the problems here

REFERENCES

[I] ATKINSONA On the Measurement of Inequality Journal of Economic Theory 2 (1970) 244-263

[2] BECKERG Human Capital 2nd Edition New York National Bureau of Economic Research 1975

(31 BECKER G AND N TOMES An Equilibrium Theory of the Distribution of Income and Intergenerational Mobility Journal of Political Economy 87 (1979) 1153-1 189

[4] BOUDONR Mathematical Structures of Social Mobiliiy Amsterdam Elsevier Scientific Publish- ing Co 1973

[S] BROCKW AND L MIRMANOptimal Economic Growth and Uncertainty The Discounted Case Journal of Economic Theory 4 (1972) 479-513

INTERGENERATIONAL TRANSFERS 867

[6] CHAMPERNOWNE Model of Economic Journal 63 (1953) D G A Income Distribution 318-351

[7] CONLISKJ Can Equalization of Opportunity Reduce Social Mobility American Economic Review 64 (1974) 80-90

[S] DATCHERL The Effect of Family and Community Background on the Education and Earnings of Black and White Men Unpublished Manuscript Institute for Social Research University of Michigan 1980

[9] DIAMONDP National Debt in a Neoclassical Growth Model American Economic Review 55 (1965) 1126-1 150

[lo] DUNCAN O D FEATHERMAN AND B DUNCAN S O C ~ O ~ C O ~ O ~ ~ CBackground and Achievement New York Seminar Press 1972

[ l l ] FRIEDMANA Foundations of Modern Analysis New York Holt Rinehart and Winston Inc 1970

[12] FuTIA C A Stochastic Approach to Economic Dynamics Bell Telephone Laboratories Discussion Paper 1976

[13] GOLDBERGERA S Models and Methods in the IQ Debate Part I SSRI Workshop Series No 7710 University of Wisconsin 1977

[14] HARSANYIJ C Cardinal Welfare Individualistic Ethics and Interpersonal Comparisons of Utility Journal of Political Economy 63 (1955) 309-321

[IS] JENCKSC ET ALInequality A Reassessment of the Effect of Family and Schooling in America New York Harper and Row 1972

[16] KOHLBERGE A Model of Economic Growth with Altruism between Generations Journal of Economic Theory 13 (1976) 1-13

[17] KOOPMANST Representation of Preference Orderings over Time in Decisions and Organiza- tion ed by C B McGuire and R Radner Amsterdam North-Holland Publishing Co 1972

[IS] LOURYG C Essays in the Theory of the Distribution of Income PhD Thesis Massachu- setts Institute of Technology 1976

[19] MANDELBROTB Stable Paretian Random Functions and the Multiplicative Variation of Income Econometrica 29 (1961) 517-543

[20] MINCERJ Investment in Human Capital and Personal Income Distribution Journal of Political Economy 66 (1958) 281-302

[211 -- The Distribution of Labor Incomes A Survey Journal of Economic Literature 8 (1970) 1-26

[22] OKUNAEquity and Eficiency The Big Tradeoff Washington D C The Brookings Institu- tion 1975

1231 RAWL~J A Theory of Justice Cambridge Massachusetts Harvard University Press 1971 24 ROTHSCHILD M AND J STIGLITZ Increasing Risk I A Definition Journal of Economic

Theory 2 (1970) 225-243 [25] ROY A D The Distribution of Earnings and Individual Output Economic Journal 60

(1950) 489-505 [26] SAMUELSON An Exact Consumption-Loan Model of Interest with or without the Social P

Contrivance of Money Journal of Political Economy 66 (1958) 467-482 [27] SHORROCKSA The Measurement of Mobility Econometrica 46 (1978) 1013-1024 [28] SOLOW R On the Dynamics of the Income Distribution PhD Thesis Harvard University

1951 1291 VARIAN H Redistributive Taxation as Social Insurance Journal o f Public Economics 14

(1980) 49-68 [30] ZECKHAUSERR Risk Spreading and Distribution Chapter 8 in Redistribution Through Public

Choice ed by Hochman and Peterson New York Columbia University Press 1974

You have printed the following article

Intergenerational Transfers and the Distribution of EarningsGlenn C LouryEconometrica Vol 49 No 4 (Jul 1981) pp 843-867Stable URL

httplinksjstororgsicisici=0012-96822819810729493A43C8433AITATDO3E20CO3B2-C

This article references the following linked citations If you are trying to access articles from anoff-campus location you may be required to first logon via your library web site to access JSTOR Pleasevisit your librarys website or contact a librarian to learn about options for remote access to JSTOR

[Footnotes]

4 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

9 National Debt in a Neoclassical Growth ModelPeter A DiamondThe American Economic Review Vol 55 No 5 Part 1 (Dec 1965) pp 1126-1150Stable URL

httplinksjstororgsicisici=0002-82822819651229553A53C11263ANDIANG3E20CO3B2-L

References

3 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

httpwwwjstororg

LINKED CITATIONS- Page 1 of 3 -

NOTE The reference numbering from the original has been maintained in this citation list

860 GLENN C LOURY

we use the following results reported in the excellent recent survey of Futia

PROPOSITION(Futia [12 Theorems 36 37 46 52 and 561) Under Assump- tions A1-A3 the process represented by equations (5) and (6) has at least one invariant measure v E9and aIways converges to some invariant measure More- over if in addition A4 and the foIIowing condition hold then the invariant measure is unique

CONDITION1 There exists a yo E [O 81 with the property that for every integer k gt 1 any number y E [0 y] and any neighborhood U of yo one can find an integer n such that P ~ ( ~ U ) gt 0

The m-step transition probability P m ( y A ) is defined inductively by P l ( y ~ ) r P ( y A ) and

Thus we will have proven the theorem if we can exhibit an earnings level yo every neighborhood of which is with positive probability and finite periodicity entered infinitely often from any initial earnings level We will need the following notation

The following Lemmata will also be useful

PROOF Implement the change of variables a = h - ( x e(y)) in (6) QED

REMARKThus our transition probability on the bounded state space [0 y] arises from integrating a bounded measurable density function It is this fact which enables us to employ Futias results indicated in the Proposition above

LEMMA2 Let S m ( y ) be the support of the probabiIity P m ( y ) E9Then S m ( y ) = (x(y)x(y))

PROOF Define

= 0 otherwise

INTERGENERATIONAL TRANSFERS

and inductively

Now by Lemma 1 we have

Thus Sm(y) = x I pm(y x )gt O) and we need only show that pe(yx) gt 0 iff x E (xr (y) x(y)) This is done by induction Obviously pl(x y) gt 0 iff x E (x(~) xi(y)) from the definitions Suppose the hypothesis holds for pm- (Y x) and define Am(x y) -- z E [0 co) ( P ( ~ Z )~ - (Z x) gt 0) Now pn(y x) gt 0 iff p(Am(x y)) gt 0 ~ u t gt 0 iff z E (X(Y)X(Y)) whilepm-(zx) P ( ~ Z ) gt 0 iff x E (x-(t)x-(z)) It follows from Assumptions A2(i) and A4 that the functions x(-) and x() are strictly increasing Thus

So p(Am(x y)) gt 0 iff both (x-I)-(x) lt and (x-I)-(x) gt xi(y) that is iff x(y) gt x gt x(y) QED

LEMMA3 Lim xom(y)= 0 Vy E [0 71 Moreover there exists y gt 0 such that lim inf x(y) gt y Vy E [0 J]

PROOF For y gt 0 by A2(i) and Theorem 1 xi(y) lt e(y) lty Thus the sequence of numbers x(y)) is monotonically decreasing and bounded below by zero Let x = lirn x(y) Then if g gt0 we have by continuity of xi(-)

a contradiction The second statement in the lemma follows from the fact that x(O) gt 0 (by A2(i)) so by continuity there exists y gt 0 such that x ] ( ~ ) gty Vy lt 9 Thus x(y)) could never have a convergent subsequence approaching a value less than y QED

We can now complete the argument for Condition 1 By Lemma 2 Pm(yA) gt 0 if p(A n (x(y) x(y))) gt 0 By Lemma 3 there is an interval (O) such that [A C (0 y)] [ A c (x(y) x(y))] with y given and m sufficiently large Thus for any point yo E (0 y) we have shown that every neighborhood of yo has positive probability of being entered after a finite number of steps from any original pointy E [0 oo) Moreover we have shown that Pm(yA) gt Opmk(y A) gt 0 for k gt 1 and A c (0 y) Condition 1 then follows

It remains to argue that the support of the invariant measure is connected and compact But this follows from A2(iii) and A4 which imply the exi3tence of a smallest point F E [0 oo) with the property that x ] ( ~ ) lty Vy gtF Thus when

862 GLENN C LOURY

y gtJthe supremum of the support of Pm(y ) monotonically decreases until it falls below JHence the invariant measure has support contained in the interval [Oy=] It is clear that the supremum of the support of the invariant measure y satisfies y = Moreover limmx(y) 0 V y implies that if y is in the ~ ( ~ 3 =

support of the invariant measure and y lty then y is in the support Hence the support is connected QED

PROOFOF THEOREM3 For any t E12the map T C [0y]+C [0y] is defined by

T ( y )= max E U(c ( t (h (a y - c) y - c)) ) y E [ O J]o lt c lt y

Inductively define T= T ( T - I ) n = 23 By Theorem 1 we know there is a unique y E C[Oy] such that lim I(T)(y)- V(y)( = 0 V y E [0J] Now let ( y )= U(yO)and suppose t + t Then we have

(T )(Y)= oppyEm U ( t ( h ( f f Y - c) Y - c)O))

lt max E U(c ~ ( 1 ( h ( a - c) y - c) 0 ) ) o lt c lt y

by thf facts that the composition of increasing concave functions is concave and that t + t implies the expectation of any concave function of after tax income is no less under t than under t Suppose inductively that ( T - ) ( ~ ) lt ( T - I )

( y ) V y E LO 71 Then

= max (T- )(t(h(a - c) y - c ) ) )E ~ ( c o lt c lt y

4 max E U ( C ( T P 1 ) ( t ( h ( a - c) y - c ) ) )o lt c lt y

lt man (T ) ( i ( h ( a - c) y - c ) ) )E ~ ( c o lt c lt y

where the first inequality follows from the induction hypothesis and monotonic- ity of U while the second inequality is established by the argument employed just above Hence we have shown that (T)(y)4 (Tp)(y )by gt 0 V n = 1 2 It then follows from pointwise convergence that V(y) lt VP(y) V y E

[O JI QED

PROOFOF THEOREM4 Let 7 be the average income in the laissez faire equilibrium Denote by x the earnings of an arbitrary mature individual and by

863 INTERGENERATIONAL TRANSFERS

y the earnings of his parent in the preceding period In what follows we will think of (x y) as a jointly random vector Now it is a well known fact of bivariate distribution theory that

Moreover the variance of income under public education is given by

var = var(x ( y = x)

The proof proceeds by noting that A5 implies var(x I y) is a convex function of y One may then apply Jensens inequality which along with (14) implies the result

Consider

Thus the conditional variance of offsprings earnings increases (decreases) with parental income iff ability and education are complements (substitutes) Further- more it follows from (14) that var(x) gt var(x ( y = x) when a2haa ae = 0 Now one may calculate

This term is nonnegative under A5 It follows that var(x) gt var Moreover the concavity of h(a e(y)) in y also implies x lt jflf(a)h(a e(x))p (da) so mean income is no lower under public education QED

864 GLENN C LOURY

PROOFOF THEOREM5 Fubinis Theorem implies that in equilibrium

Now v invariant implies

l o I I f ( a ) [ j 0[O 00) (h(a e(y)) - y ) v ( d y ) ] y(da) =

which simply states that per capita earnings is constant across a generation By A2(i) the bracketed term in the integrand above is strictly increasing in a Thus 36 E (0l) such that

f (a )J ( h ( a e(y)) -y)v (dy) 20 as a 1 6 10 a)

Hence

We now outline an argument showing that the economy neednt be strongly meritocratic Consider the following

where ( 2 3 is the random vector of offspring-parent earnings and y- (a x) is defined by h ( a e ( y - ( a ~ ) ) )r x That is an individual observed to have earnings x will have an innate endowment exceeding a if and only if that individuals parent had an income less than yP (a x) Let p (y x ) be as defined in the proof of Lemma 2 under the proof of Theorem 2 given above Then we have that

Thus the equilibrium distribution is meritocratic in the strong sense if and only if the right-hand side above is an increasing function of x Notice that this

865 INTERGENERATIONAL TRANSFERS

restriction is in principle testable when the distribution of parental income conditional on offsprings income can be observed

An increase in x has two effects First it leads to an increase in y- (a x) the critical level below which parental income must lie if 6 is to exceed a However an increase in x also shifts to the right the conditional distribution of parents earnings (Ylx) tending to reduce the right-hand side above That this latter effect can outweigh the former may be seen by the following (admittedly extreme) example Suppose there are only two possible levels of training invest- ment which parents can make q and 2 with q lt 2 Imagine further that -x = h(g 1) lt h(P 0) = X Now imagine observing two individuals one of whom earns 2 + r the other of whom earns x - E where E is a small positive number Then it must be the case (by continuity of h( )) that the person with earnings X + E has ability close to zero while the person who earns x- r has ability close to one QED

7 CONCLUSIONS

The analysis presented here brings together two separate strands of the literature on economic inequality On the one hand we have modelled the income dynamics as a stochastic process and studied the properties of its ergodic distribution This approach has a long history (for example [6 7 19 25 281) On the other hand the underlying structure of this stochastic process results from maximizing decisions of the economic agents As such it has much in common with the traditional human capital approach to earnings [2 3 20 211 Yet we have departed from the human capital school in one crucial respect-the assumption that a perfect market for educational loans exists We have assumed instead a balkanized market where each family must generate internally funds for the training of its young We believe this assumption parallels more closely actual circumstances As a consequence of this assumption family background exercises an independent constraining effect on social mobility Our analysis is in this regard consistent with the approach to the study of mobility which economists and sociologists have found so fruitful in recent years (for example [4 8 10 151)

The present approach commends itself in one important additional respect By grounding training decisions in rational choice we are able to rigorously analyze some normative issues regarding public policies which would alter the distribu- tion of income Moreover this analysis [Section 41 has not presupposed the existence of some social welfare function Rather it has proceeded on a purely individualistic basis We have seen that redistributive policies can serve to improve both the allocation of risks and of training resources when capital markets are incomplete Indeed normative prescriptions similar in spirit to those of Harsanyi [14] or Rawls [23] to the extent that they are derived from an analysis of the risks which an individual faces in some ex ante original position are here given a positive behavioral grounding We have replaced the

866 GLENN C LOURY

veil of ignorance with the generation gap across which parents cannot look to perceive the position which their offspring will occupy in society

Finally we should note some shortcomings of the foregoing analysis By far the strongest assumption we have made is that innate abilities are independent across generations While the exact nature of intergenerational correlations has not yet been satisfactorily ascertained the existence of some such effects is impossible to deny Theorems 1 and 2 survive with slight modification general- ization to a Markovian structure on abilities However the tax analysis of Section 4 is lost once such allowance is made This seems a stimulating topic for further investigation

Another weak point is the fact that the welfare implications of income taxation studied in Section 4 are limited to education-specific tax systems These may not be politically or administratively feasible and do not involve redistribution across social classes In any case a similar analysis of tax systems dependent only upon income would be desirable This appears to be a very difficult matter in general Moreover a strong result like that given in Theorem 3 will no longer hold for such schemes

Finally the framework developed here suggests an interesting research prob- lem in the normative economics of social mobility The transition probability for a society P ( y A ) contains all relevant information regarding both cross-section inequality in the long run (if the process is ergodic) as well as intergenerational mobility One might consider axiom systems to impose on orderings of such transition functions which reflect intuitive notions of what it means to say that one social structure is more equal than another The question then becomes when can orderings satisfying these intuitive axioms be represented in the simple way suggested in equation (S) (That is when may social structures P be ranked by the expected value of some indirect utility function under their ergodic distribution)

Universizy of Michigan

Manuscript received July 1978 revision received June 1980

I3see Goldberger [13] for an informative discussion of the problems here

REFERENCES

[I] ATKINSONA On the Measurement of Inequality Journal of Economic Theory 2 (1970) 244-263

[2] BECKERG Human Capital 2nd Edition New York National Bureau of Economic Research 1975

(31 BECKER G AND N TOMES An Equilibrium Theory of the Distribution of Income and Intergenerational Mobility Journal of Political Economy 87 (1979) 1153-1 189

[4] BOUDONR Mathematical Structures of Social Mobiliiy Amsterdam Elsevier Scientific Publish- ing Co 1973

[S] BROCKW AND L MIRMANOptimal Economic Growth and Uncertainty The Discounted Case Journal of Economic Theory 4 (1972) 479-513

INTERGENERATIONAL TRANSFERS 867

[6] CHAMPERNOWNE Model of Economic Journal 63 (1953) D G A Income Distribution 318-351

[7] CONLISKJ Can Equalization of Opportunity Reduce Social Mobility American Economic Review 64 (1974) 80-90

[S] DATCHERL The Effect of Family and Community Background on the Education and Earnings of Black and White Men Unpublished Manuscript Institute for Social Research University of Michigan 1980

[9] DIAMONDP National Debt in a Neoclassical Growth Model American Economic Review 55 (1965) 1126-1 150

[lo] DUNCAN O D FEATHERMAN AND B DUNCAN S O C ~ O ~ C O ~ O ~ ~ CBackground and Achievement New York Seminar Press 1972

[ l l ] FRIEDMANA Foundations of Modern Analysis New York Holt Rinehart and Winston Inc 1970

[12] FuTIA C A Stochastic Approach to Economic Dynamics Bell Telephone Laboratories Discussion Paper 1976

[13] GOLDBERGERA S Models and Methods in the IQ Debate Part I SSRI Workshop Series No 7710 University of Wisconsin 1977

[14] HARSANYIJ C Cardinal Welfare Individualistic Ethics and Interpersonal Comparisons of Utility Journal of Political Economy 63 (1955) 309-321

[IS] JENCKSC ET ALInequality A Reassessment of the Effect of Family and Schooling in America New York Harper and Row 1972

[16] KOHLBERGE A Model of Economic Growth with Altruism between Generations Journal of Economic Theory 13 (1976) 1-13

[17] KOOPMANST Representation of Preference Orderings over Time in Decisions and Organiza- tion ed by C B McGuire and R Radner Amsterdam North-Holland Publishing Co 1972

[IS] LOURYG C Essays in the Theory of the Distribution of Income PhD Thesis Massachu- setts Institute of Technology 1976

[19] MANDELBROTB Stable Paretian Random Functions and the Multiplicative Variation of Income Econometrica 29 (1961) 517-543

[20] MINCERJ Investment in Human Capital and Personal Income Distribution Journal of Political Economy 66 (1958) 281-302

[211 -- The Distribution of Labor Incomes A Survey Journal of Economic Literature 8 (1970) 1-26

[22] OKUNAEquity and Eficiency The Big Tradeoff Washington D C The Brookings Institu- tion 1975

1231 RAWL~J A Theory of Justice Cambridge Massachusetts Harvard University Press 1971 24 ROTHSCHILD M AND J STIGLITZ Increasing Risk I A Definition Journal of Economic

Theory 2 (1970) 225-243 [25] ROY A D The Distribution of Earnings and Individual Output Economic Journal 60

(1950) 489-505 [26] SAMUELSON An Exact Consumption-Loan Model of Interest with or without the Social P

Contrivance of Money Journal of Political Economy 66 (1958) 467-482 [27] SHORROCKSA The Measurement of Mobility Econometrica 46 (1978) 1013-1024 [28] SOLOW R On the Dynamics of the Income Distribution PhD Thesis Harvard University

1951 1291 VARIAN H Redistributive Taxation as Social Insurance Journal o f Public Economics 14

(1980) 49-68 [30] ZECKHAUSERR Risk Spreading and Distribution Chapter 8 in Redistribution Through Public

Choice ed by Hochman and Peterson New York Columbia University Press 1974

You have printed the following article

Intergenerational Transfers and the Distribution of EarningsGlenn C LouryEconometrica Vol 49 No 4 (Jul 1981) pp 843-867Stable URL

httplinksjstororgsicisici=0012-96822819810729493A43C8433AITATDO3E20CO3B2-C

This article references the following linked citations If you are trying to access articles from anoff-campus location you may be required to first logon via your library web site to access JSTOR Pleasevisit your librarys website or contact a librarian to learn about options for remote access to JSTOR

[Footnotes]

4 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

9 National Debt in a Neoclassical Growth ModelPeter A DiamondThe American Economic Review Vol 55 No 5 Part 1 (Dec 1965) pp 1126-1150Stable URL

httplinksjstororgsicisici=0002-82822819651229553A53C11263ANDIANG3E20CO3B2-L

References

3 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

httpwwwjstororg

LINKED CITATIONS- Page 1 of 3 -

NOTE The reference numbering from the original has been maintained in this citation list

INTERGENERATIONAL TRANSFERS

and inductively

Now by Lemma 1 we have

Thus Sm(y) = x I pm(y x )gt O) and we need only show that pe(yx) gt 0 iff x E (xr (y) x(y)) This is done by induction Obviously pl(x y) gt 0 iff x E (x(~) xi(y)) from the definitions Suppose the hypothesis holds for pm- (Y x) and define Am(x y) -- z E [0 co) ( P ( ~ Z )~ - (Z x) gt 0) Now pn(y x) gt 0 iff p(Am(x y)) gt 0 ~ u t gt 0 iff z E (X(Y)X(Y)) whilepm-(zx) P ( ~ Z ) gt 0 iff x E (x-(t)x-(z)) It follows from Assumptions A2(i) and A4 that the functions x(-) and x() are strictly increasing Thus

So p(Am(x y)) gt 0 iff both (x-I)-(x) lt and (x-I)-(x) gt xi(y) that is iff x(y) gt x gt x(y) QED

LEMMA3 Lim xom(y)= 0 Vy E [0 71 Moreover there exists y gt 0 such that lim inf x(y) gt y Vy E [0 J]

PROOF For y gt 0 by A2(i) and Theorem 1 xi(y) lt e(y) lty Thus the sequence of numbers x(y)) is monotonically decreasing and bounded below by zero Let x = lirn x(y) Then if g gt0 we have by continuity of xi(-)

a contradiction The second statement in the lemma follows from the fact that x(O) gt 0 (by A2(i)) so by continuity there exists y gt 0 such that x ] ( ~ ) gty Vy lt 9 Thus x(y)) could never have a convergent subsequence approaching a value less than y QED

We can now complete the argument for Condition 1 By Lemma 2 Pm(yA) gt 0 if p(A n (x(y) x(y))) gt 0 By Lemma 3 there is an interval (O) such that [A C (0 y)] [ A c (x(y) x(y))] with y given and m sufficiently large Thus for any point yo E (0 y) we have shown that every neighborhood of yo has positive probability of being entered after a finite number of steps from any original pointy E [0 oo) Moreover we have shown that Pm(yA) gt Opmk(y A) gt 0 for k gt 1 and A c (0 y) Condition 1 then follows

It remains to argue that the support of the invariant measure is connected and compact But this follows from A2(iii) and A4 which imply the exi3tence of a smallest point F E [0 oo) with the property that x ] ( ~ ) lty Vy gtF Thus when

862 GLENN C LOURY

y gtJthe supremum of the support of Pm(y ) monotonically decreases until it falls below JHence the invariant measure has support contained in the interval [Oy=] It is clear that the supremum of the support of the invariant measure y satisfies y = Moreover limmx(y) 0 V y implies that if y is in the ~ ( ~ 3 =

support of the invariant measure and y lty then y is in the support Hence the support is connected QED

PROOFOF THEOREM3 For any t E12the map T C [0y]+C [0y] is defined by

T ( y )= max E U(c ( t (h (a y - c) y - c)) ) y E [ O J]o lt c lt y

Inductively define T= T ( T - I ) n = 23 By Theorem 1 we know there is a unique y E C[Oy] such that lim I(T)(y)- V(y)( = 0 V y E [0J] Now let ( y )= U(yO)and suppose t + t Then we have

(T )(Y)= oppyEm U ( t ( h ( f f Y - c) Y - c)O))

lt max E U(c ~ ( 1 ( h ( a - c) y - c) 0 ) ) o lt c lt y

by thf facts that the composition of increasing concave functions is concave and that t + t implies the expectation of any concave function of after tax income is no less under t than under t Suppose inductively that ( T - ) ( ~ ) lt ( T - I )

( y ) V y E LO 71 Then

= max (T- )(t(h(a - c) y - c ) ) )E ~ ( c o lt c lt y

4 max E U ( C ( T P 1 ) ( t ( h ( a - c) y - c ) ) )o lt c lt y

lt man (T ) ( i ( h ( a - c) y - c ) ) )E ~ ( c o lt c lt y

where the first inequality follows from the induction hypothesis and monotonic- ity of U while the second inequality is established by the argument employed just above Hence we have shown that (T)(y)4 (Tp)(y )by gt 0 V n = 1 2 It then follows from pointwise convergence that V(y) lt VP(y) V y E

[O JI QED

PROOFOF THEOREM4 Let 7 be the average income in the laissez faire equilibrium Denote by x the earnings of an arbitrary mature individual and by

863 INTERGENERATIONAL TRANSFERS

y the earnings of his parent in the preceding period In what follows we will think of (x y) as a jointly random vector Now it is a well known fact of bivariate distribution theory that

Moreover the variance of income under public education is given by

var = var(x ( y = x)

The proof proceeds by noting that A5 implies var(x I y) is a convex function of y One may then apply Jensens inequality which along with (14) implies the result

Consider

Thus the conditional variance of offsprings earnings increases (decreases) with parental income iff ability and education are complements (substitutes) Further- more it follows from (14) that var(x) gt var(x ( y = x) when a2haa ae = 0 Now one may calculate

This term is nonnegative under A5 It follows that var(x) gt var Moreover the concavity of h(a e(y)) in y also implies x lt jflf(a)h(a e(x))p (da) so mean income is no lower under public education QED

864 GLENN C LOURY

PROOFOF THEOREM5 Fubinis Theorem implies that in equilibrium

Now v invariant implies

l o I I f ( a ) [ j 0[O 00) (h(a e(y)) - y ) v ( d y ) ] y(da) =

which simply states that per capita earnings is constant across a generation By A2(i) the bracketed term in the integrand above is strictly increasing in a Thus 36 E (0l) such that

f (a )J ( h ( a e(y)) -y)v (dy) 20 as a 1 6 10 a)

Hence

We now outline an argument showing that the economy neednt be strongly meritocratic Consider the following

where ( 2 3 is the random vector of offspring-parent earnings and y- (a x) is defined by h ( a e ( y - ( a ~ ) ) )r x That is an individual observed to have earnings x will have an innate endowment exceeding a if and only if that individuals parent had an income less than yP (a x) Let p (y x ) be as defined in the proof of Lemma 2 under the proof of Theorem 2 given above Then we have that

Thus the equilibrium distribution is meritocratic in the strong sense if and only if the right-hand side above is an increasing function of x Notice that this

865 INTERGENERATIONAL TRANSFERS

restriction is in principle testable when the distribution of parental income conditional on offsprings income can be observed

An increase in x has two effects First it leads to an increase in y- (a x) the critical level below which parental income must lie if 6 is to exceed a However an increase in x also shifts to the right the conditional distribution of parents earnings (Ylx) tending to reduce the right-hand side above That this latter effect can outweigh the former may be seen by the following (admittedly extreme) example Suppose there are only two possible levels of training invest- ment which parents can make q and 2 with q lt 2 Imagine further that -x = h(g 1) lt h(P 0) = X Now imagine observing two individuals one of whom earns 2 + r the other of whom earns x - E where E is a small positive number Then it must be the case (by continuity of h( )) that the person with earnings X + E has ability close to zero while the person who earns x- r has ability close to one QED

7 CONCLUSIONS

The analysis presented here brings together two separate strands of the literature on economic inequality On the one hand we have modelled the income dynamics as a stochastic process and studied the properties of its ergodic distribution This approach has a long history (for example [6 7 19 25 281) On the other hand the underlying structure of this stochastic process results from maximizing decisions of the economic agents As such it has much in common with the traditional human capital approach to earnings [2 3 20 211 Yet we have departed from the human capital school in one crucial respect-the assumption that a perfect market for educational loans exists We have assumed instead a balkanized market where each family must generate internally funds for the training of its young We believe this assumption parallels more closely actual circumstances As a consequence of this assumption family background exercises an independent constraining effect on social mobility Our analysis is in this regard consistent with the approach to the study of mobility which economists and sociologists have found so fruitful in recent years (for example [4 8 10 151)

The present approach commends itself in one important additional respect By grounding training decisions in rational choice we are able to rigorously analyze some normative issues regarding public policies which would alter the distribu- tion of income Moreover this analysis [Section 41 has not presupposed the existence of some social welfare function Rather it has proceeded on a purely individualistic basis We have seen that redistributive policies can serve to improve both the allocation of risks and of training resources when capital markets are incomplete Indeed normative prescriptions similar in spirit to those of Harsanyi [14] or Rawls [23] to the extent that they are derived from an analysis of the risks which an individual faces in some ex ante original position are here given a positive behavioral grounding We have replaced the

866 GLENN C LOURY

veil of ignorance with the generation gap across which parents cannot look to perceive the position which their offspring will occupy in society

Finally we should note some shortcomings of the foregoing analysis By far the strongest assumption we have made is that innate abilities are independent across generations While the exact nature of intergenerational correlations has not yet been satisfactorily ascertained the existence of some such effects is impossible to deny Theorems 1 and 2 survive with slight modification general- ization to a Markovian structure on abilities However the tax analysis of Section 4 is lost once such allowance is made This seems a stimulating topic for further investigation

Another weak point is the fact that the welfare implications of income taxation studied in Section 4 are limited to education-specific tax systems These may not be politically or administratively feasible and do not involve redistribution across social classes In any case a similar analysis of tax systems dependent only upon income would be desirable This appears to be a very difficult matter in general Moreover a strong result like that given in Theorem 3 will no longer hold for such schemes

Finally the framework developed here suggests an interesting research prob- lem in the normative economics of social mobility The transition probability for a society P ( y A ) contains all relevant information regarding both cross-section inequality in the long run (if the process is ergodic) as well as intergenerational mobility One might consider axiom systems to impose on orderings of such transition functions which reflect intuitive notions of what it means to say that one social structure is more equal than another The question then becomes when can orderings satisfying these intuitive axioms be represented in the simple way suggested in equation (S) (That is when may social structures P be ranked by the expected value of some indirect utility function under their ergodic distribution)

Universizy of Michigan

Manuscript received July 1978 revision received June 1980

I3see Goldberger [13] for an informative discussion of the problems here

REFERENCES

[I] ATKINSONA On the Measurement of Inequality Journal of Economic Theory 2 (1970) 244-263

[2] BECKERG Human Capital 2nd Edition New York National Bureau of Economic Research 1975

(31 BECKER G AND N TOMES An Equilibrium Theory of the Distribution of Income and Intergenerational Mobility Journal of Political Economy 87 (1979) 1153-1 189

[4] BOUDONR Mathematical Structures of Social Mobiliiy Amsterdam Elsevier Scientific Publish- ing Co 1973

[S] BROCKW AND L MIRMANOptimal Economic Growth and Uncertainty The Discounted Case Journal of Economic Theory 4 (1972) 479-513

INTERGENERATIONAL TRANSFERS 867

[6] CHAMPERNOWNE Model of Economic Journal 63 (1953) D G A Income Distribution 318-351

[7] CONLISKJ Can Equalization of Opportunity Reduce Social Mobility American Economic Review 64 (1974) 80-90

[S] DATCHERL The Effect of Family and Community Background on the Education and Earnings of Black and White Men Unpublished Manuscript Institute for Social Research University of Michigan 1980

[9] DIAMONDP National Debt in a Neoclassical Growth Model American Economic Review 55 (1965) 1126-1 150

[lo] DUNCAN O D FEATHERMAN AND B DUNCAN S O C ~ O ~ C O ~ O ~ ~ CBackground and Achievement New York Seminar Press 1972

[ l l ] FRIEDMANA Foundations of Modern Analysis New York Holt Rinehart and Winston Inc 1970

[12] FuTIA C A Stochastic Approach to Economic Dynamics Bell Telephone Laboratories Discussion Paper 1976

[13] GOLDBERGERA S Models and Methods in the IQ Debate Part I SSRI Workshop Series No 7710 University of Wisconsin 1977

[14] HARSANYIJ C Cardinal Welfare Individualistic Ethics and Interpersonal Comparisons of Utility Journal of Political Economy 63 (1955) 309-321

[IS] JENCKSC ET ALInequality A Reassessment of the Effect of Family and Schooling in America New York Harper and Row 1972

[16] KOHLBERGE A Model of Economic Growth with Altruism between Generations Journal of Economic Theory 13 (1976) 1-13

[17] KOOPMANST Representation of Preference Orderings over Time in Decisions and Organiza- tion ed by C B McGuire and R Radner Amsterdam North-Holland Publishing Co 1972

[IS] LOURYG C Essays in the Theory of the Distribution of Income PhD Thesis Massachu- setts Institute of Technology 1976

[19] MANDELBROTB Stable Paretian Random Functions and the Multiplicative Variation of Income Econometrica 29 (1961) 517-543

[20] MINCERJ Investment in Human Capital and Personal Income Distribution Journal of Political Economy 66 (1958) 281-302

[211 -- The Distribution of Labor Incomes A Survey Journal of Economic Literature 8 (1970) 1-26

[22] OKUNAEquity and Eficiency The Big Tradeoff Washington D C The Brookings Institu- tion 1975

1231 RAWL~J A Theory of Justice Cambridge Massachusetts Harvard University Press 1971 24 ROTHSCHILD M AND J STIGLITZ Increasing Risk I A Definition Journal of Economic

Theory 2 (1970) 225-243 [25] ROY A D The Distribution of Earnings and Individual Output Economic Journal 60

(1950) 489-505 [26] SAMUELSON An Exact Consumption-Loan Model of Interest with or without the Social P

Contrivance of Money Journal of Political Economy 66 (1958) 467-482 [27] SHORROCKSA The Measurement of Mobility Econometrica 46 (1978) 1013-1024 [28] SOLOW R On the Dynamics of the Income Distribution PhD Thesis Harvard University

1951 1291 VARIAN H Redistributive Taxation as Social Insurance Journal o f Public Economics 14

(1980) 49-68 [30] ZECKHAUSERR Risk Spreading and Distribution Chapter 8 in Redistribution Through Public

Choice ed by Hochman and Peterson New York Columbia University Press 1974

You have printed the following article

Intergenerational Transfers and the Distribution of EarningsGlenn C LouryEconometrica Vol 49 No 4 (Jul 1981) pp 843-867Stable URL

httplinksjstororgsicisici=0012-96822819810729493A43C8433AITATDO3E20CO3B2-C

This article references the following linked citations If you are trying to access articles from anoff-campus location you may be required to first logon via your library web site to access JSTOR Pleasevisit your librarys website or contact a librarian to learn about options for remote access to JSTOR

[Footnotes]

4 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

9 National Debt in a Neoclassical Growth ModelPeter A DiamondThe American Economic Review Vol 55 No 5 Part 1 (Dec 1965) pp 1126-1150Stable URL

httplinksjstororgsicisici=0002-82822819651229553A53C11263ANDIANG3E20CO3B2-L

References

3 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

httpwwwjstororg

LINKED CITATIONS- Page 1 of 3 -

NOTE The reference numbering from the original has been maintained in this citation list

862 GLENN C LOURY

y gtJthe supremum of the support of Pm(y ) monotonically decreases until it falls below JHence the invariant measure has support contained in the interval [Oy=] It is clear that the supremum of the support of the invariant measure y satisfies y = Moreover limmx(y) 0 V y implies that if y is in the ~ ( ~ 3 =

support of the invariant measure and y lty then y is in the support Hence the support is connected QED

PROOFOF THEOREM3 For any t E12the map T C [0y]+C [0y] is defined by

T ( y )= max E U(c ( t (h (a y - c) y - c)) ) y E [ O J]o lt c lt y

Inductively define T= T ( T - I ) n = 23 By Theorem 1 we know there is a unique y E C[Oy] such that lim I(T)(y)- V(y)( = 0 V y E [0J] Now let ( y )= U(yO)and suppose t + t Then we have

(T )(Y)= oppyEm U ( t ( h ( f f Y - c) Y - c)O))

lt max E U(c ~ ( 1 ( h ( a - c) y - c) 0 ) ) o lt c lt y

by thf facts that the composition of increasing concave functions is concave and that t + t implies the expectation of any concave function of after tax income is no less under t than under t Suppose inductively that ( T - ) ( ~ ) lt ( T - I )

( y ) V y E LO 71 Then

= max (T- )(t(h(a - c) y - c ) ) )E ~ ( c o lt c lt y

4 max E U ( C ( T P 1 ) ( t ( h ( a - c) y - c ) ) )o lt c lt y

lt man (T ) ( i ( h ( a - c) y - c ) ) )E ~ ( c o lt c lt y

where the first inequality follows from the induction hypothesis and monotonic- ity of U while the second inequality is established by the argument employed just above Hence we have shown that (T)(y)4 (Tp)(y )by gt 0 V n = 1 2 It then follows from pointwise convergence that V(y) lt VP(y) V y E

[O JI QED

PROOFOF THEOREM4 Let 7 be the average income in the laissez faire equilibrium Denote by x the earnings of an arbitrary mature individual and by

863 INTERGENERATIONAL TRANSFERS

y the earnings of his parent in the preceding period In what follows we will think of (x y) as a jointly random vector Now it is a well known fact of bivariate distribution theory that

Moreover the variance of income under public education is given by

var = var(x ( y = x)

The proof proceeds by noting that A5 implies var(x I y) is a convex function of y One may then apply Jensens inequality which along with (14) implies the result

Consider

Thus the conditional variance of offsprings earnings increases (decreases) with parental income iff ability and education are complements (substitutes) Further- more it follows from (14) that var(x) gt var(x ( y = x) when a2haa ae = 0 Now one may calculate

This term is nonnegative under A5 It follows that var(x) gt var Moreover the concavity of h(a e(y)) in y also implies x lt jflf(a)h(a e(x))p (da) so mean income is no lower under public education QED

864 GLENN C LOURY

PROOFOF THEOREM5 Fubinis Theorem implies that in equilibrium

Now v invariant implies

l o I I f ( a ) [ j 0[O 00) (h(a e(y)) - y ) v ( d y ) ] y(da) =

which simply states that per capita earnings is constant across a generation By A2(i) the bracketed term in the integrand above is strictly increasing in a Thus 36 E (0l) such that

f (a )J ( h ( a e(y)) -y)v (dy) 20 as a 1 6 10 a)

Hence

We now outline an argument showing that the economy neednt be strongly meritocratic Consider the following

where ( 2 3 is the random vector of offspring-parent earnings and y- (a x) is defined by h ( a e ( y - ( a ~ ) ) )r x That is an individual observed to have earnings x will have an innate endowment exceeding a if and only if that individuals parent had an income less than yP (a x) Let p (y x ) be as defined in the proof of Lemma 2 under the proof of Theorem 2 given above Then we have that

Thus the equilibrium distribution is meritocratic in the strong sense if and only if the right-hand side above is an increasing function of x Notice that this

865 INTERGENERATIONAL TRANSFERS

restriction is in principle testable when the distribution of parental income conditional on offsprings income can be observed

An increase in x has two effects First it leads to an increase in y- (a x) the critical level below which parental income must lie if 6 is to exceed a However an increase in x also shifts to the right the conditional distribution of parents earnings (Ylx) tending to reduce the right-hand side above That this latter effect can outweigh the former may be seen by the following (admittedly extreme) example Suppose there are only two possible levels of training invest- ment which parents can make q and 2 with q lt 2 Imagine further that -x = h(g 1) lt h(P 0) = X Now imagine observing two individuals one of whom earns 2 + r the other of whom earns x - E where E is a small positive number Then it must be the case (by continuity of h( )) that the person with earnings X + E has ability close to zero while the person who earns x- r has ability close to one QED

7 CONCLUSIONS

The analysis presented here brings together two separate strands of the literature on economic inequality On the one hand we have modelled the income dynamics as a stochastic process and studied the properties of its ergodic distribution This approach has a long history (for example [6 7 19 25 281) On the other hand the underlying structure of this stochastic process results from maximizing decisions of the economic agents As such it has much in common with the traditional human capital approach to earnings [2 3 20 211 Yet we have departed from the human capital school in one crucial respect-the assumption that a perfect market for educational loans exists We have assumed instead a balkanized market where each family must generate internally funds for the training of its young We believe this assumption parallels more closely actual circumstances As a consequence of this assumption family background exercises an independent constraining effect on social mobility Our analysis is in this regard consistent with the approach to the study of mobility which economists and sociologists have found so fruitful in recent years (for example [4 8 10 151)

The present approach commends itself in one important additional respect By grounding training decisions in rational choice we are able to rigorously analyze some normative issues regarding public policies which would alter the distribu- tion of income Moreover this analysis [Section 41 has not presupposed the existence of some social welfare function Rather it has proceeded on a purely individualistic basis We have seen that redistributive policies can serve to improve both the allocation of risks and of training resources when capital markets are incomplete Indeed normative prescriptions similar in spirit to those of Harsanyi [14] or Rawls [23] to the extent that they are derived from an analysis of the risks which an individual faces in some ex ante original position are here given a positive behavioral grounding We have replaced the

866 GLENN C LOURY

veil of ignorance with the generation gap across which parents cannot look to perceive the position which their offspring will occupy in society

Finally we should note some shortcomings of the foregoing analysis By far the strongest assumption we have made is that innate abilities are independent across generations While the exact nature of intergenerational correlations has not yet been satisfactorily ascertained the existence of some such effects is impossible to deny Theorems 1 and 2 survive with slight modification general- ization to a Markovian structure on abilities However the tax analysis of Section 4 is lost once such allowance is made This seems a stimulating topic for further investigation

Another weak point is the fact that the welfare implications of income taxation studied in Section 4 are limited to education-specific tax systems These may not be politically or administratively feasible and do not involve redistribution across social classes In any case a similar analysis of tax systems dependent only upon income would be desirable This appears to be a very difficult matter in general Moreover a strong result like that given in Theorem 3 will no longer hold for such schemes

Finally the framework developed here suggests an interesting research prob- lem in the normative economics of social mobility The transition probability for a society P ( y A ) contains all relevant information regarding both cross-section inequality in the long run (if the process is ergodic) as well as intergenerational mobility One might consider axiom systems to impose on orderings of such transition functions which reflect intuitive notions of what it means to say that one social structure is more equal than another The question then becomes when can orderings satisfying these intuitive axioms be represented in the simple way suggested in equation (S) (That is when may social structures P be ranked by the expected value of some indirect utility function under their ergodic distribution)

Universizy of Michigan

Manuscript received July 1978 revision received June 1980

I3see Goldberger [13] for an informative discussion of the problems here

REFERENCES

[I] ATKINSONA On the Measurement of Inequality Journal of Economic Theory 2 (1970) 244-263

[2] BECKERG Human Capital 2nd Edition New York National Bureau of Economic Research 1975

(31 BECKER G AND N TOMES An Equilibrium Theory of the Distribution of Income and Intergenerational Mobility Journal of Political Economy 87 (1979) 1153-1 189

[4] BOUDONR Mathematical Structures of Social Mobiliiy Amsterdam Elsevier Scientific Publish- ing Co 1973

[S] BROCKW AND L MIRMANOptimal Economic Growth and Uncertainty The Discounted Case Journal of Economic Theory 4 (1972) 479-513

INTERGENERATIONAL TRANSFERS 867

[6] CHAMPERNOWNE Model of Economic Journal 63 (1953) D G A Income Distribution 318-351

[7] CONLISKJ Can Equalization of Opportunity Reduce Social Mobility American Economic Review 64 (1974) 80-90

[S] DATCHERL The Effect of Family and Community Background on the Education and Earnings of Black and White Men Unpublished Manuscript Institute for Social Research University of Michigan 1980

[9] DIAMONDP National Debt in a Neoclassical Growth Model American Economic Review 55 (1965) 1126-1 150

[lo] DUNCAN O D FEATHERMAN AND B DUNCAN S O C ~ O ~ C O ~ O ~ ~ CBackground and Achievement New York Seminar Press 1972

[ l l ] FRIEDMANA Foundations of Modern Analysis New York Holt Rinehart and Winston Inc 1970

[12] FuTIA C A Stochastic Approach to Economic Dynamics Bell Telephone Laboratories Discussion Paper 1976

[13] GOLDBERGERA S Models and Methods in the IQ Debate Part I SSRI Workshop Series No 7710 University of Wisconsin 1977

[14] HARSANYIJ C Cardinal Welfare Individualistic Ethics and Interpersonal Comparisons of Utility Journal of Political Economy 63 (1955) 309-321

[IS] JENCKSC ET ALInequality A Reassessment of the Effect of Family and Schooling in America New York Harper and Row 1972

[16] KOHLBERGE A Model of Economic Growth with Altruism between Generations Journal of Economic Theory 13 (1976) 1-13

[17] KOOPMANST Representation of Preference Orderings over Time in Decisions and Organiza- tion ed by C B McGuire and R Radner Amsterdam North-Holland Publishing Co 1972

[IS] LOURYG C Essays in the Theory of the Distribution of Income PhD Thesis Massachu- setts Institute of Technology 1976

[19] MANDELBROTB Stable Paretian Random Functions and the Multiplicative Variation of Income Econometrica 29 (1961) 517-543

[20] MINCERJ Investment in Human Capital and Personal Income Distribution Journal of Political Economy 66 (1958) 281-302

[211 -- The Distribution of Labor Incomes A Survey Journal of Economic Literature 8 (1970) 1-26

[22] OKUNAEquity and Eficiency The Big Tradeoff Washington D C The Brookings Institu- tion 1975

1231 RAWL~J A Theory of Justice Cambridge Massachusetts Harvard University Press 1971 24 ROTHSCHILD M AND J STIGLITZ Increasing Risk I A Definition Journal of Economic

Theory 2 (1970) 225-243 [25] ROY A D The Distribution of Earnings and Individual Output Economic Journal 60

(1950) 489-505 [26] SAMUELSON An Exact Consumption-Loan Model of Interest with or without the Social P

Contrivance of Money Journal of Political Economy 66 (1958) 467-482 [27] SHORROCKSA The Measurement of Mobility Econometrica 46 (1978) 1013-1024 [28] SOLOW R On the Dynamics of the Income Distribution PhD Thesis Harvard University

1951 1291 VARIAN H Redistributive Taxation as Social Insurance Journal o f Public Economics 14

(1980) 49-68 [30] ZECKHAUSERR Risk Spreading and Distribution Chapter 8 in Redistribution Through Public

Choice ed by Hochman and Peterson New York Columbia University Press 1974

You have printed the following article

Intergenerational Transfers and the Distribution of EarningsGlenn C LouryEconometrica Vol 49 No 4 (Jul 1981) pp 843-867Stable URL

httplinksjstororgsicisici=0012-96822819810729493A43C8433AITATDO3E20CO3B2-C

This article references the following linked citations If you are trying to access articles from anoff-campus location you may be required to first logon via your library web site to access JSTOR Pleasevisit your librarys website or contact a librarian to learn about options for remote access to JSTOR

[Footnotes]

4 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

9 National Debt in a Neoclassical Growth ModelPeter A DiamondThe American Economic Review Vol 55 No 5 Part 1 (Dec 1965) pp 1126-1150Stable URL

httplinksjstororgsicisici=0002-82822819651229553A53C11263ANDIANG3E20CO3B2-L

References

3 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

httpwwwjstororg

LINKED CITATIONS- Page 1 of 3 -

NOTE The reference numbering from the original has been maintained in this citation list

863 INTERGENERATIONAL TRANSFERS

y the earnings of his parent in the preceding period In what follows we will think of (x y) as a jointly random vector Now it is a well known fact of bivariate distribution theory that

Moreover the variance of income under public education is given by

var = var(x ( y = x)

The proof proceeds by noting that A5 implies var(x I y) is a convex function of y One may then apply Jensens inequality which along with (14) implies the result

Consider

Thus the conditional variance of offsprings earnings increases (decreases) with parental income iff ability and education are complements (substitutes) Further- more it follows from (14) that var(x) gt var(x ( y = x) when a2haa ae = 0 Now one may calculate

This term is nonnegative under A5 It follows that var(x) gt var Moreover the concavity of h(a e(y)) in y also implies x lt jflf(a)h(a e(x))p (da) so mean income is no lower under public education QED

864 GLENN C LOURY

PROOFOF THEOREM5 Fubinis Theorem implies that in equilibrium

Now v invariant implies

l o I I f ( a ) [ j 0[O 00) (h(a e(y)) - y ) v ( d y ) ] y(da) =

which simply states that per capita earnings is constant across a generation By A2(i) the bracketed term in the integrand above is strictly increasing in a Thus 36 E (0l) such that

f (a )J ( h ( a e(y)) -y)v (dy) 20 as a 1 6 10 a)

Hence

We now outline an argument showing that the economy neednt be strongly meritocratic Consider the following

where ( 2 3 is the random vector of offspring-parent earnings and y- (a x) is defined by h ( a e ( y - ( a ~ ) ) )r x That is an individual observed to have earnings x will have an innate endowment exceeding a if and only if that individuals parent had an income less than yP (a x) Let p (y x ) be as defined in the proof of Lemma 2 under the proof of Theorem 2 given above Then we have that

Thus the equilibrium distribution is meritocratic in the strong sense if and only if the right-hand side above is an increasing function of x Notice that this

865 INTERGENERATIONAL TRANSFERS

restriction is in principle testable when the distribution of parental income conditional on offsprings income can be observed

An increase in x has two effects First it leads to an increase in y- (a x) the critical level below which parental income must lie if 6 is to exceed a However an increase in x also shifts to the right the conditional distribution of parents earnings (Ylx) tending to reduce the right-hand side above That this latter effect can outweigh the former may be seen by the following (admittedly extreme) example Suppose there are only two possible levels of training invest- ment which parents can make q and 2 with q lt 2 Imagine further that -x = h(g 1) lt h(P 0) = X Now imagine observing two individuals one of whom earns 2 + r the other of whom earns x - E where E is a small positive number Then it must be the case (by continuity of h( )) that the person with earnings X + E has ability close to zero while the person who earns x- r has ability close to one QED

7 CONCLUSIONS

The analysis presented here brings together two separate strands of the literature on economic inequality On the one hand we have modelled the income dynamics as a stochastic process and studied the properties of its ergodic distribution This approach has a long history (for example [6 7 19 25 281) On the other hand the underlying structure of this stochastic process results from maximizing decisions of the economic agents As such it has much in common with the traditional human capital approach to earnings [2 3 20 211 Yet we have departed from the human capital school in one crucial respect-the assumption that a perfect market for educational loans exists We have assumed instead a balkanized market where each family must generate internally funds for the training of its young We believe this assumption parallels more closely actual circumstances As a consequence of this assumption family background exercises an independent constraining effect on social mobility Our analysis is in this regard consistent with the approach to the study of mobility which economists and sociologists have found so fruitful in recent years (for example [4 8 10 151)

The present approach commends itself in one important additional respect By grounding training decisions in rational choice we are able to rigorously analyze some normative issues regarding public policies which would alter the distribu- tion of income Moreover this analysis [Section 41 has not presupposed the existence of some social welfare function Rather it has proceeded on a purely individualistic basis We have seen that redistributive policies can serve to improve both the allocation of risks and of training resources when capital markets are incomplete Indeed normative prescriptions similar in spirit to those of Harsanyi [14] or Rawls [23] to the extent that they are derived from an analysis of the risks which an individual faces in some ex ante original position are here given a positive behavioral grounding We have replaced the

866 GLENN C LOURY

veil of ignorance with the generation gap across which parents cannot look to perceive the position which their offspring will occupy in society

Finally we should note some shortcomings of the foregoing analysis By far the strongest assumption we have made is that innate abilities are independent across generations While the exact nature of intergenerational correlations has not yet been satisfactorily ascertained the existence of some such effects is impossible to deny Theorems 1 and 2 survive with slight modification general- ization to a Markovian structure on abilities However the tax analysis of Section 4 is lost once such allowance is made This seems a stimulating topic for further investigation

Another weak point is the fact that the welfare implications of income taxation studied in Section 4 are limited to education-specific tax systems These may not be politically or administratively feasible and do not involve redistribution across social classes In any case a similar analysis of tax systems dependent only upon income would be desirable This appears to be a very difficult matter in general Moreover a strong result like that given in Theorem 3 will no longer hold for such schemes

Finally the framework developed here suggests an interesting research prob- lem in the normative economics of social mobility The transition probability for a society P ( y A ) contains all relevant information regarding both cross-section inequality in the long run (if the process is ergodic) as well as intergenerational mobility One might consider axiom systems to impose on orderings of such transition functions which reflect intuitive notions of what it means to say that one social structure is more equal than another The question then becomes when can orderings satisfying these intuitive axioms be represented in the simple way suggested in equation (S) (That is when may social structures P be ranked by the expected value of some indirect utility function under their ergodic distribution)

Universizy of Michigan

Manuscript received July 1978 revision received June 1980

I3see Goldberger [13] for an informative discussion of the problems here

REFERENCES

[I] ATKINSONA On the Measurement of Inequality Journal of Economic Theory 2 (1970) 244-263

[2] BECKERG Human Capital 2nd Edition New York National Bureau of Economic Research 1975

(31 BECKER G AND N TOMES An Equilibrium Theory of the Distribution of Income and Intergenerational Mobility Journal of Political Economy 87 (1979) 1153-1 189

[4] BOUDONR Mathematical Structures of Social Mobiliiy Amsterdam Elsevier Scientific Publish- ing Co 1973

[S] BROCKW AND L MIRMANOptimal Economic Growth and Uncertainty The Discounted Case Journal of Economic Theory 4 (1972) 479-513

INTERGENERATIONAL TRANSFERS 867

[6] CHAMPERNOWNE Model of Economic Journal 63 (1953) D G A Income Distribution 318-351

[7] CONLISKJ Can Equalization of Opportunity Reduce Social Mobility American Economic Review 64 (1974) 80-90

[S] DATCHERL The Effect of Family and Community Background on the Education and Earnings of Black and White Men Unpublished Manuscript Institute for Social Research University of Michigan 1980

[9] DIAMONDP National Debt in a Neoclassical Growth Model American Economic Review 55 (1965) 1126-1 150

[lo] DUNCAN O D FEATHERMAN AND B DUNCAN S O C ~ O ~ C O ~ O ~ ~ CBackground and Achievement New York Seminar Press 1972

[ l l ] FRIEDMANA Foundations of Modern Analysis New York Holt Rinehart and Winston Inc 1970

[12] FuTIA C A Stochastic Approach to Economic Dynamics Bell Telephone Laboratories Discussion Paper 1976

[13] GOLDBERGERA S Models and Methods in the IQ Debate Part I SSRI Workshop Series No 7710 University of Wisconsin 1977

[14] HARSANYIJ C Cardinal Welfare Individualistic Ethics and Interpersonal Comparisons of Utility Journal of Political Economy 63 (1955) 309-321

[IS] JENCKSC ET ALInequality A Reassessment of the Effect of Family and Schooling in America New York Harper and Row 1972

[16] KOHLBERGE A Model of Economic Growth with Altruism between Generations Journal of Economic Theory 13 (1976) 1-13

[17] KOOPMANST Representation of Preference Orderings over Time in Decisions and Organiza- tion ed by C B McGuire and R Radner Amsterdam North-Holland Publishing Co 1972

[IS] LOURYG C Essays in the Theory of the Distribution of Income PhD Thesis Massachu- setts Institute of Technology 1976

[19] MANDELBROTB Stable Paretian Random Functions and the Multiplicative Variation of Income Econometrica 29 (1961) 517-543

[20] MINCERJ Investment in Human Capital and Personal Income Distribution Journal of Political Economy 66 (1958) 281-302

[211 -- The Distribution of Labor Incomes A Survey Journal of Economic Literature 8 (1970) 1-26

[22] OKUNAEquity and Eficiency The Big Tradeoff Washington D C The Brookings Institu- tion 1975

1231 RAWL~J A Theory of Justice Cambridge Massachusetts Harvard University Press 1971 24 ROTHSCHILD M AND J STIGLITZ Increasing Risk I A Definition Journal of Economic

Theory 2 (1970) 225-243 [25] ROY A D The Distribution of Earnings and Individual Output Economic Journal 60

(1950) 489-505 [26] SAMUELSON An Exact Consumption-Loan Model of Interest with or without the Social P

Contrivance of Money Journal of Political Economy 66 (1958) 467-482 [27] SHORROCKSA The Measurement of Mobility Econometrica 46 (1978) 1013-1024 [28] SOLOW R On the Dynamics of the Income Distribution PhD Thesis Harvard University

1951 1291 VARIAN H Redistributive Taxation as Social Insurance Journal o f Public Economics 14

(1980) 49-68 [30] ZECKHAUSERR Risk Spreading and Distribution Chapter 8 in Redistribution Through Public

Choice ed by Hochman and Peterson New York Columbia University Press 1974

You have printed the following article

Intergenerational Transfers and the Distribution of EarningsGlenn C LouryEconometrica Vol 49 No 4 (Jul 1981) pp 843-867Stable URL

httplinksjstororgsicisici=0012-96822819810729493A43C8433AITATDO3E20CO3B2-C

This article references the following linked citations If you are trying to access articles from anoff-campus location you may be required to first logon via your library web site to access JSTOR Pleasevisit your librarys website or contact a librarian to learn about options for remote access to JSTOR

[Footnotes]

4 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

9 National Debt in a Neoclassical Growth ModelPeter A DiamondThe American Economic Review Vol 55 No 5 Part 1 (Dec 1965) pp 1126-1150Stable URL

httplinksjstororgsicisici=0002-82822819651229553A53C11263ANDIANG3E20CO3B2-L

References

3 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

httpwwwjstororg

LINKED CITATIONS- Page 1 of 3 -

NOTE The reference numbering from the original has been maintained in this citation list

864 GLENN C LOURY

PROOFOF THEOREM5 Fubinis Theorem implies that in equilibrium

Now v invariant implies

l o I I f ( a ) [ j 0[O 00) (h(a e(y)) - y ) v ( d y ) ] y(da) =

which simply states that per capita earnings is constant across a generation By A2(i) the bracketed term in the integrand above is strictly increasing in a Thus 36 E (0l) such that

f (a )J ( h ( a e(y)) -y)v (dy) 20 as a 1 6 10 a)

Hence

We now outline an argument showing that the economy neednt be strongly meritocratic Consider the following

where ( 2 3 is the random vector of offspring-parent earnings and y- (a x) is defined by h ( a e ( y - ( a ~ ) ) )r x That is an individual observed to have earnings x will have an innate endowment exceeding a if and only if that individuals parent had an income less than yP (a x) Let p (y x ) be as defined in the proof of Lemma 2 under the proof of Theorem 2 given above Then we have that

Thus the equilibrium distribution is meritocratic in the strong sense if and only if the right-hand side above is an increasing function of x Notice that this

865 INTERGENERATIONAL TRANSFERS

restriction is in principle testable when the distribution of parental income conditional on offsprings income can be observed

An increase in x has two effects First it leads to an increase in y- (a x) the critical level below which parental income must lie if 6 is to exceed a However an increase in x also shifts to the right the conditional distribution of parents earnings (Ylx) tending to reduce the right-hand side above That this latter effect can outweigh the former may be seen by the following (admittedly extreme) example Suppose there are only two possible levels of training invest- ment which parents can make q and 2 with q lt 2 Imagine further that -x = h(g 1) lt h(P 0) = X Now imagine observing two individuals one of whom earns 2 + r the other of whom earns x - E where E is a small positive number Then it must be the case (by continuity of h( )) that the person with earnings X + E has ability close to zero while the person who earns x- r has ability close to one QED

7 CONCLUSIONS

The analysis presented here brings together two separate strands of the literature on economic inequality On the one hand we have modelled the income dynamics as a stochastic process and studied the properties of its ergodic distribution This approach has a long history (for example [6 7 19 25 281) On the other hand the underlying structure of this stochastic process results from maximizing decisions of the economic agents As such it has much in common with the traditional human capital approach to earnings [2 3 20 211 Yet we have departed from the human capital school in one crucial respect-the assumption that a perfect market for educational loans exists We have assumed instead a balkanized market where each family must generate internally funds for the training of its young We believe this assumption parallels more closely actual circumstances As a consequence of this assumption family background exercises an independent constraining effect on social mobility Our analysis is in this regard consistent with the approach to the study of mobility which economists and sociologists have found so fruitful in recent years (for example [4 8 10 151)

The present approach commends itself in one important additional respect By grounding training decisions in rational choice we are able to rigorously analyze some normative issues regarding public policies which would alter the distribu- tion of income Moreover this analysis [Section 41 has not presupposed the existence of some social welfare function Rather it has proceeded on a purely individualistic basis We have seen that redistributive policies can serve to improve both the allocation of risks and of training resources when capital markets are incomplete Indeed normative prescriptions similar in spirit to those of Harsanyi [14] or Rawls [23] to the extent that they are derived from an analysis of the risks which an individual faces in some ex ante original position are here given a positive behavioral grounding We have replaced the

866 GLENN C LOURY

veil of ignorance with the generation gap across which parents cannot look to perceive the position which their offspring will occupy in society

Finally we should note some shortcomings of the foregoing analysis By far the strongest assumption we have made is that innate abilities are independent across generations While the exact nature of intergenerational correlations has not yet been satisfactorily ascertained the existence of some such effects is impossible to deny Theorems 1 and 2 survive with slight modification general- ization to a Markovian structure on abilities However the tax analysis of Section 4 is lost once such allowance is made This seems a stimulating topic for further investigation

Another weak point is the fact that the welfare implications of income taxation studied in Section 4 are limited to education-specific tax systems These may not be politically or administratively feasible and do not involve redistribution across social classes In any case a similar analysis of tax systems dependent only upon income would be desirable This appears to be a very difficult matter in general Moreover a strong result like that given in Theorem 3 will no longer hold for such schemes

Finally the framework developed here suggests an interesting research prob- lem in the normative economics of social mobility The transition probability for a society P ( y A ) contains all relevant information regarding both cross-section inequality in the long run (if the process is ergodic) as well as intergenerational mobility One might consider axiom systems to impose on orderings of such transition functions which reflect intuitive notions of what it means to say that one social structure is more equal than another The question then becomes when can orderings satisfying these intuitive axioms be represented in the simple way suggested in equation (S) (That is when may social structures P be ranked by the expected value of some indirect utility function under their ergodic distribution)

Universizy of Michigan

Manuscript received July 1978 revision received June 1980

I3see Goldberger [13] for an informative discussion of the problems here

REFERENCES

[I] ATKINSONA On the Measurement of Inequality Journal of Economic Theory 2 (1970) 244-263

[2] BECKERG Human Capital 2nd Edition New York National Bureau of Economic Research 1975

(31 BECKER G AND N TOMES An Equilibrium Theory of the Distribution of Income and Intergenerational Mobility Journal of Political Economy 87 (1979) 1153-1 189

[4] BOUDONR Mathematical Structures of Social Mobiliiy Amsterdam Elsevier Scientific Publish- ing Co 1973

[S] BROCKW AND L MIRMANOptimal Economic Growth and Uncertainty The Discounted Case Journal of Economic Theory 4 (1972) 479-513

INTERGENERATIONAL TRANSFERS 867

[6] CHAMPERNOWNE Model of Economic Journal 63 (1953) D G A Income Distribution 318-351

[7] CONLISKJ Can Equalization of Opportunity Reduce Social Mobility American Economic Review 64 (1974) 80-90

[S] DATCHERL The Effect of Family and Community Background on the Education and Earnings of Black and White Men Unpublished Manuscript Institute for Social Research University of Michigan 1980

[9] DIAMONDP National Debt in a Neoclassical Growth Model American Economic Review 55 (1965) 1126-1 150

[lo] DUNCAN O D FEATHERMAN AND B DUNCAN S O C ~ O ~ C O ~ O ~ ~ CBackground and Achievement New York Seminar Press 1972

[ l l ] FRIEDMANA Foundations of Modern Analysis New York Holt Rinehart and Winston Inc 1970

[12] FuTIA C A Stochastic Approach to Economic Dynamics Bell Telephone Laboratories Discussion Paper 1976

[13] GOLDBERGERA S Models and Methods in the IQ Debate Part I SSRI Workshop Series No 7710 University of Wisconsin 1977

[14] HARSANYIJ C Cardinal Welfare Individualistic Ethics and Interpersonal Comparisons of Utility Journal of Political Economy 63 (1955) 309-321

[IS] JENCKSC ET ALInequality A Reassessment of the Effect of Family and Schooling in America New York Harper and Row 1972

[16] KOHLBERGE A Model of Economic Growth with Altruism between Generations Journal of Economic Theory 13 (1976) 1-13

[17] KOOPMANST Representation of Preference Orderings over Time in Decisions and Organiza- tion ed by C B McGuire and R Radner Amsterdam North-Holland Publishing Co 1972

[IS] LOURYG C Essays in the Theory of the Distribution of Income PhD Thesis Massachu- setts Institute of Technology 1976

[19] MANDELBROTB Stable Paretian Random Functions and the Multiplicative Variation of Income Econometrica 29 (1961) 517-543

[20] MINCERJ Investment in Human Capital and Personal Income Distribution Journal of Political Economy 66 (1958) 281-302

[211 -- The Distribution of Labor Incomes A Survey Journal of Economic Literature 8 (1970) 1-26

[22] OKUNAEquity and Eficiency The Big Tradeoff Washington D C The Brookings Institu- tion 1975

1231 RAWL~J A Theory of Justice Cambridge Massachusetts Harvard University Press 1971 24 ROTHSCHILD M AND J STIGLITZ Increasing Risk I A Definition Journal of Economic

Theory 2 (1970) 225-243 [25] ROY A D The Distribution of Earnings and Individual Output Economic Journal 60

(1950) 489-505 [26] SAMUELSON An Exact Consumption-Loan Model of Interest with or without the Social P

Contrivance of Money Journal of Political Economy 66 (1958) 467-482 [27] SHORROCKSA The Measurement of Mobility Econometrica 46 (1978) 1013-1024 [28] SOLOW R On the Dynamics of the Income Distribution PhD Thesis Harvard University

1951 1291 VARIAN H Redistributive Taxation as Social Insurance Journal o f Public Economics 14

(1980) 49-68 [30] ZECKHAUSERR Risk Spreading and Distribution Chapter 8 in Redistribution Through Public

Choice ed by Hochman and Peterson New York Columbia University Press 1974

You have printed the following article

Intergenerational Transfers and the Distribution of EarningsGlenn C LouryEconometrica Vol 49 No 4 (Jul 1981) pp 843-867Stable URL

httplinksjstororgsicisici=0012-96822819810729493A43C8433AITATDO3E20CO3B2-C

This article references the following linked citations If you are trying to access articles from anoff-campus location you may be required to first logon via your library web site to access JSTOR Pleasevisit your librarys website or contact a librarian to learn about options for remote access to JSTOR

[Footnotes]

4 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

9 National Debt in a Neoclassical Growth ModelPeter A DiamondThe American Economic Review Vol 55 No 5 Part 1 (Dec 1965) pp 1126-1150Stable URL

httplinksjstororgsicisici=0002-82822819651229553A53C11263ANDIANG3E20CO3B2-L

References

3 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

httpwwwjstororg

LINKED CITATIONS- Page 1 of 3 -

NOTE The reference numbering from the original has been maintained in this citation list

865 INTERGENERATIONAL TRANSFERS

restriction is in principle testable when the distribution of parental income conditional on offsprings income can be observed

An increase in x has two effects First it leads to an increase in y- (a x) the critical level below which parental income must lie if 6 is to exceed a However an increase in x also shifts to the right the conditional distribution of parents earnings (Ylx) tending to reduce the right-hand side above That this latter effect can outweigh the former may be seen by the following (admittedly extreme) example Suppose there are only two possible levels of training invest- ment which parents can make q and 2 with q lt 2 Imagine further that -x = h(g 1) lt h(P 0) = X Now imagine observing two individuals one of whom earns 2 + r the other of whom earns x - E where E is a small positive number Then it must be the case (by continuity of h( )) that the person with earnings X + E has ability close to zero while the person who earns x- r has ability close to one QED

7 CONCLUSIONS

The analysis presented here brings together two separate strands of the literature on economic inequality On the one hand we have modelled the income dynamics as a stochastic process and studied the properties of its ergodic distribution This approach has a long history (for example [6 7 19 25 281) On the other hand the underlying structure of this stochastic process results from maximizing decisions of the economic agents As such it has much in common with the traditional human capital approach to earnings [2 3 20 211 Yet we have departed from the human capital school in one crucial respect-the assumption that a perfect market for educational loans exists We have assumed instead a balkanized market where each family must generate internally funds for the training of its young We believe this assumption parallels more closely actual circumstances As a consequence of this assumption family background exercises an independent constraining effect on social mobility Our analysis is in this regard consistent with the approach to the study of mobility which economists and sociologists have found so fruitful in recent years (for example [4 8 10 151)

The present approach commends itself in one important additional respect By grounding training decisions in rational choice we are able to rigorously analyze some normative issues regarding public policies which would alter the distribu- tion of income Moreover this analysis [Section 41 has not presupposed the existence of some social welfare function Rather it has proceeded on a purely individualistic basis We have seen that redistributive policies can serve to improve both the allocation of risks and of training resources when capital markets are incomplete Indeed normative prescriptions similar in spirit to those of Harsanyi [14] or Rawls [23] to the extent that they are derived from an analysis of the risks which an individual faces in some ex ante original position are here given a positive behavioral grounding We have replaced the

866 GLENN C LOURY

veil of ignorance with the generation gap across which parents cannot look to perceive the position which their offspring will occupy in society

Finally we should note some shortcomings of the foregoing analysis By far the strongest assumption we have made is that innate abilities are independent across generations While the exact nature of intergenerational correlations has not yet been satisfactorily ascertained the existence of some such effects is impossible to deny Theorems 1 and 2 survive with slight modification general- ization to a Markovian structure on abilities However the tax analysis of Section 4 is lost once such allowance is made This seems a stimulating topic for further investigation

Another weak point is the fact that the welfare implications of income taxation studied in Section 4 are limited to education-specific tax systems These may not be politically or administratively feasible and do not involve redistribution across social classes In any case a similar analysis of tax systems dependent only upon income would be desirable This appears to be a very difficult matter in general Moreover a strong result like that given in Theorem 3 will no longer hold for such schemes

Finally the framework developed here suggests an interesting research prob- lem in the normative economics of social mobility The transition probability for a society P ( y A ) contains all relevant information regarding both cross-section inequality in the long run (if the process is ergodic) as well as intergenerational mobility One might consider axiom systems to impose on orderings of such transition functions which reflect intuitive notions of what it means to say that one social structure is more equal than another The question then becomes when can orderings satisfying these intuitive axioms be represented in the simple way suggested in equation (S) (That is when may social structures P be ranked by the expected value of some indirect utility function under their ergodic distribution)

Universizy of Michigan

Manuscript received July 1978 revision received June 1980

I3see Goldberger [13] for an informative discussion of the problems here

REFERENCES

[I] ATKINSONA On the Measurement of Inequality Journal of Economic Theory 2 (1970) 244-263

[2] BECKERG Human Capital 2nd Edition New York National Bureau of Economic Research 1975

(31 BECKER G AND N TOMES An Equilibrium Theory of the Distribution of Income and Intergenerational Mobility Journal of Political Economy 87 (1979) 1153-1 189

[4] BOUDONR Mathematical Structures of Social Mobiliiy Amsterdam Elsevier Scientific Publish- ing Co 1973

[S] BROCKW AND L MIRMANOptimal Economic Growth and Uncertainty The Discounted Case Journal of Economic Theory 4 (1972) 479-513

INTERGENERATIONAL TRANSFERS 867

[6] CHAMPERNOWNE Model of Economic Journal 63 (1953) D G A Income Distribution 318-351

[7] CONLISKJ Can Equalization of Opportunity Reduce Social Mobility American Economic Review 64 (1974) 80-90

[S] DATCHERL The Effect of Family and Community Background on the Education and Earnings of Black and White Men Unpublished Manuscript Institute for Social Research University of Michigan 1980

[9] DIAMONDP National Debt in a Neoclassical Growth Model American Economic Review 55 (1965) 1126-1 150

[lo] DUNCAN O D FEATHERMAN AND B DUNCAN S O C ~ O ~ C O ~ O ~ ~ CBackground and Achievement New York Seminar Press 1972

[ l l ] FRIEDMANA Foundations of Modern Analysis New York Holt Rinehart and Winston Inc 1970

[12] FuTIA C A Stochastic Approach to Economic Dynamics Bell Telephone Laboratories Discussion Paper 1976

[13] GOLDBERGERA S Models and Methods in the IQ Debate Part I SSRI Workshop Series No 7710 University of Wisconsin 1977

[14] HARSANYIJ C Cardinal Welfare Individualistic Ethics and Interpersonal Comparisons of Utility Journal of Political Economy 63 (1955) 309-321

[IS] JENCKSC ET ALInequality A Reassessment of the Effect of Family and Schooling in America New York Harper and Row 1972

[16] KOHLBERGE A Model of Economic Growth with Altruism between Generations Journal of Economic Theory 13 (1976) 1-13

[17] KOOPMANST Representation of Preference Orderings over Time in Decisions and Organiza- tion ed by C B McGuire and R Radner Amsterdam North-Holland Publishing Co 1972

[IS] LOURYG C Essays in the Theory of the Distribution of Income PhD Thesis Massachu- setts Institute of Technology 1976

[19] MANDELBROTB Stable Paretian Random Functions and the Multiplicative Variation of Income Econometrica 29 (1961) 517-543

[20] MINCERJ Investment in Human Capital and Personal Income Distribution Journal of Political Economy 66 (1958) 281-302

[211 -- The Distribution of Labor Incomes A Survey Journal of Economic Literature 8 (1970) 1-26

[22] OKUNAEquity and Eficiency The Big Tradeoff Washington D C The Brookings Institu- tion 1975

1231 RAWL~J A Theory of Justice Cambridge Massachusetts Harvard University Press 1971 24 ROTHSCHILD M AND J STIGLITZ Increasing Risk I A Definition Journal of Economic

Theory 2 (1970) 225-243 [25] ROY A D The Distribution of Earnings and Individual Output Economic Journal 60

(1950) 489-505 [26] SAMUELSON An Exact Consumption-Loan Model of Interest with or without the Social P

Contrivance of Money Journal of Political Economy 66 (1958) 467-482 [27] SHORROCKSA The Measurement of Mobility Econometrica 46 (1978) 1013-1024 [28] SOLOW R On the Dynamics of the Income Distribution PhD Thesis Harvard University

1951 1291 VARIAN H Redistributive Taxation as Social Insurance Journal o f Public Economics 14

(1980) 49-68 [30] ZECKHAUSERR Risk Spreading and Distribution Chapter 8 in Redistribution Through Public

Choice ed by Hochman and Peterson New York Columbia University Press 1974

You have printed the following article

Intergenerational Transfers and the Distribution of EarningsGlenn C LouryEconometrica Vol 49 No 4 (Jul 1981) pp 843-867Stable URL

httplinksjstororgsicisici=0012-96822819810729493A43C8433AITATDO3E20CO3B2-C

This article references the following linked citations If you are trying to access articles from anoff-campus location you may be required to first logon via your library web site to access JSTOR Pleasevisit your librarys website or contact a librarian to learn about options for remote access to JSTOR

[Footnotes]

4 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

9 National Debt in a Neoclassical Growth ModelPeter A DiamondThe American Economic Review Vol 55 No 5 Part 1 (Dec 1965) pp 1126-1150Stable URL

httplinksjstororgsicisici=0002-82822819651229553A53C11263ANDIANG3E20CO3B2-L

References

3 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

httpwwwjstororg

LINKED CITATIONS- Page 1 of 3 -

NOTE The reference numbering from the original has been maintained in this citation list

866 GLENN C LOURY

veil of ignorance with the generation gap across which parents cannot look to perceive the position which their offspring will occupy in society

Finally we should note some shortcomings of the foregoing analysis By far the strongest assumption we have made is that innate abilities are independent across generations While the exact nature of intergenerational correlations has not yet been satisfactorily ascertained the existence of some such effects is impossible to deny Theorems 1 and 2 survive with slight modification general- ization to a Markovian structure on abilities However the tax analysis of Section 4 is lost once such allowance is made This seems a stimulating topic for further investigation

Another weak point is the fact that the welfare implications of income taxation studied in Section 4 are limited to education-specific tax systems These may not be politically or administratively feasible and do not involve redistribution across social classes In any case a similar analysis of tax systems dependent only upon income would be desirable This appears to be a very difficult matter in general Moreover a strong result like that given in Theorem 3 will no longer hold for such schemes

Finally the framework developed here suggests an interesting research prob- lem in the normative economics of social mobility The transition probability for a society P ( y A ) contains all relevant information regarding both cross-section inequality in the long run (if the process is ergodic) as well as intergenerational mobility One might consider axiom systems to impose on orderings of such transition functions which reflect intuitive notions of what it means to say that one social structure is more equal than another The question then becomes when can orderings satisfying these intuitive axioms be represented in the simple way suggested in equation (S) (That is when may social structures P be ranked by the expected value of some indirect utility function under their ergodic distribution)

Universizy of Michigan

Manuscript received July 1978 revision received June 1980

I3see Goldberger [13] for an informative discussion of the problems here

REFERENCES

[I] ATKINSONA On the Measurement of Inequality Journal of Economic Theory 2 (1970) 244-263

[2] BECKERG Human Capital 2nd Edition New York National Bureau of Economic Research 1975

(31 BECKER G AND N TOMES An Equilibrium Theory of the Distribution of Income and Intergenerational Mobility Journal of Political Economy 87 (1979) 1153-1 189

[4] BOUDONR Mathematical Structures of Social Mobiliiy Amsterdam Elsevier Scientific Publish- ing Co 1973

[S] BROCKW AND L MIRMANOptimal Economic Growth and Uncertainty The Discounted Case Journal of Economic Theory 4 (1972) 479-513

INTERGENERATIONAL TRANSFERS 867

[6] CHAMPERNOWNE Model of Economic Journal 63 (1953) D G A Income Distribution 318-351

[7] CONLISKJ Can Equalization of Opportunity Reduce Social Mobility American Economic Review 64 (1974) 80-90

[S] DATCHERL The Effect of Family and Community Background on the Education and Earnings of Black and White Men Unpublished Manuscript Institute for Social Research University of Michigan 1980

[9] DIAMONDP National Debt in a Neoclassical Growth Model American Economic Review 55 (1965) 1126-1 150

[lo] DUNCAN O D FEATHERMAN AND B DUNCAN S O C ~ O ~ C O ~ O ~ ~ CBackground and Achievement New York Seminar Press 1972

[ l l ] FRIEDMANA Foundations of Modern Analysis New York Holt Rinehart and Winston Inc 1970

[12] FuTIA C A Stochastic Approach to Economic Dynamics Bell Telephone Laboratories Discussion Paper 1976

[13] GOLDBERGERA S Models and Methods in the IQ Debate Part I SSRI Workshop Series No 7710 University of Wisconsin 1977

[14] HARSANYIJ C Cardinal Welfare Individualistic Ethics and Interpersonal Comparisons of Utility Journal of Political Economy 63 (1955) 309-321

[IS] JENCKSC ET ALInequality A Reassessment of the Effect of Family and Schooling in America New York Harper and Row 1972

[16] KOHLBERGE A Model of Economic Growth with Altruism between Generations Journal of Economic Theory 13 (1976) 1-13

[17] KOOPMANST Representation of Preference Orderings over Time in Decisions and Organiza- tion ed by C B McGuire and R Radner Amsterdam North-Holland Publishing Co 1972

[IS] LOURYG C Essays in the Theory of the Distribution of Income PhD Thesis Massachu- setts Institute of Technology 1976

[19] MANDELBROTB Stable Paretian Random Functions and the Multiplicative Variation of Income Econometrica 29 (1961) 517-543

[20] MINCERJ Investment in Human Capital and Personal Income Distribution Journal of Political Economy 66 (1958) 281-302

[211 -- The Distribution of Labor Incomes A Survey Journal of Economic Literature 8 (1970) 1-26

[22] OKUNAEquity and Eficiency The Big Tradeoff Washington D C The Brookings Institu- tion 1975

1231 RAWL~J A Theory of Justice Cambridge Massachusetts Harvard University Press 1971 24 ROTHSCHILD M AND J STIGLITZ Increasing Risk I A Definition Journal of Economic

Theory 2 (1970) 225-243 [25] ROY A D The Distribution of Earnings and Individual Output Economic Journal 60

(1950) 489-505 [26] SAMUELSON An Exact Consumption-Loan Model of Interest with or without the Social P

Contrivance of Money Journal of Political Economy 66 (1958) 467-482 [27] SHORROCKSA The Measurement of Mobility Econometrica 46 (1978) 1013-1024 [28] SOLOW R On the Dynamics of the Income Distribution PhD Thesis Harvard University

1951 1291 VARIAN H Redistributive Taxation as Social Insurance Journal o f Public Economics 14

(1980) 49-68 [30] ZECKHAUSERR Risk Spreading and Distribution Chapter 8 in Redistribution Through Public

Choice ed by Hochman and Peterson New York Columbia University Press 1974

You have printed the following article

Intergenerational Transfers and the Distribution of EarningsGlenn C LouryEconometrica Vol 49 No 4 (Jul 1981) pp 843-867Stable URL

httplinksjstororgsicisici=0012-96822819810729493A43C8433AITATDO3E20CO3B2-C

This article references the following linked citations If you are trying to access articles from anoff-campus location you may be required to first logon via your library web site to access JSTOR Pleasevisit your librarys website or contact a librarian to learn about options for remote access to JSTOR

[Footnotes]

4 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

9 National Debt in a Neoclassical Growth ModelPeter A DiamondThe American Economic Review Vol 55 No 5 Part 1 (Dec 1965) pp 1126-1150Stable URL

httplinksjstororgsicisici=0002-82822819651229553A53C11263ANDIANG3E20CO3B2-L

References

3 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

httpwwwjstororg

LINKED CITATIONS- Page 1 of 3 -

NOTE The reference numbering from the original has been maintained in this citation list

INTERGENERATIONAL TRANSFERS 867

[6] CHAMPERNOWNE Model of Economic Journal 63 (1953) D G A Income Distribution 318-351

[7] CONLISKJ Can Equalization of Opportunity Reduce Social Mobility American Economic Review 64 (1974) 80-90

[S] DATCHERL The Effect of Family and Community Background on the Education and Earnings of Black and White Men Unpublished Manuscript Institute for Social Research University of Michigan 1980

[9] DIAMONDP National Debt in a Neoclassical Growth Model American Economic Review 55 (1965) 1126-1 150

[lo] DUNCAN O D FEATHERMAN AND B DUNCAN S O C ~ O ~ C O ~ O ~ ~ CBackground and Achievement New York Seminar Press 1972

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Intergenerational Transfers and the Distribution of EarningsGlenn C LouryEconometrica Vol 49 No 4 (Jul 1981) pp 843-867Stable URL

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[Footnotes]

4 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

9 National Debt in a Neoclassical Growth ModelPeter A DiamondThe American Economic Review Vol 55 No 5 Part 1 (Dec 1965) pp 1126-1150Stable URL

httplinksjstororgsicisici=0002-82822819651229553A53C11263ANDIANG3E20CO3B2-L

References

3 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

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You have printed the following article

Intergenerational Transfers and the Distribution of EarningsGlenn C LouryEconometrica Vol 49 No 4 (Jul 1981) pp 843-867Stable URL

httplinksjstororgsicisici=0012-96822819810729493A43C8433AITATDO3E20CO3B2-C

This article references the following linked citations If you are trying to access articles from anoff-campus location you may be required to first logon via your library web site to access JSTOR Pleasevisit your librarys website or contact a librarian to learn about options for remote access to JSTOR

[Footnotes]

4 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

9 National Debt in a Neoclassical Growth ModelPeter A DiamondThe American Economic Review Vol 55 No 5 Part 1 (Dec 1965) pp 1126-1150Stable URL

httplinksjstororgsicisici=0002-82822819651229553A53C11263ANDIANG3E20CO3B2-L

References

3 An Equilibrium Theory of the Distribution of Income and Intergenerational MobilityGary S Becker Nigel TomesThe Journal of Political Economy Vol 87 No 6 (Dec 1979) pp 1153-1189Stable URL

httplinksjstororgsicisici=0022-38082819791229873A63C11533AAETOTD3E20CO3B2-1

httpwwwjstororg

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6 A Model of Income DistributionD G ChampernowneThe Economic Journal Vol 63 No 250 (Jun 1953) pp 318-351Stable URL

httplinksjstororgsicisici=0013-01332819530629633A2503C3183AAMOID3E20CO3B2-N

7 Can Equalization of Opportunity Reduce Social MobilityJohn ConliskThe American Economic Review Vol 64 No 1 (Mar 1974) pp 80-90Stable URL

httplinksjstororgsicisici=0002-82822819740329643A13C803ACEOORS3E20CO3B2-L

9 National Debt in a Neoclassical Growth ModelPeter A DiamondThe American Economic Review Vol 55 No 5 Part 1 (Dec 1965) pp 1126-1150Stable URL

httplinksjstororgsicisici=0002-82822819651229553A53C11263ANDIANG3E20CO3B2-L

14 Cardinal Welfare Individualistic Ethics and Interpersonal Comparisons of UtilityJohn C HarsanyiThe Journal of Political Economy Vol 63 No 4 (Aug 1955) pp 309-321Stable URL

httplinksjstororgsicisici=0022-38082819550829633A43C3093ACWIEAI3E20CO3B2-P

19 Stable Paretian Random Functions and the Multiplicative Variation of IncomeBenoit MandelbrotEconometrica Vol 29 No 4 (Oct 1961) pp 517-543Stable URL

httplinksjstororgsicisici=0012-96822819611029293A43C5173ASPRFAT3E20CO3B2-Q

20 Investment in Human Capital and Personal Income DistributionJacob MincerThe Journal of Political Economy Vol 66 No 4 (Aug 1958) pp 281-302Stable URL

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21 The Distribution of Labor Incomes A Survey With Special Reference to the Human CapitalApproachJacob MincerJournal of Economic Literature Vol 8 No 1 (Mar 1970) pp 1-26Stable URL

httplinksjstororgsicisici=0022-0515281970032983A13C13ATDOLIA3E20CO3B2-2

25 The Distribution of Earnings and of Individual OutputA D RoyThe Economic Journal Vol 60 No 239 (Sep 1950) pp 489-505Stable URL

httplinksjstororgsicisici=0013-01332819500929603A2393C4893ATDOEAO3E20CO3B2-P

26 An Exact Consumption-Loan Model of Interest with or without the Social Contrivance ofMoneyPaul A SamuelsonThe Journal of Political Economy Vol 66 No 6 (Dec 1958) pp 467-482Stable URL

httplinksjstororgsicisici=0022-38082819581229663A63C4673AAECMOI3E20CO3B2-Z

27 The Measurement of MobilityA F ShorrocksEconometrica Vol 46 No 5 (Sep 1978) pp 1013-1024Stable URL

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